Unit 1-wave motion
In a long line of people waiting to buy tickets, the first person leaves and a pulse of motion occurs as people step forward to fill the gap. As each person steps forward, the gap moves through the line. Is the propagation of this gap a. transverse or b. longitudinal
longitudinal
What type of waves are sound waves?
longitudinal he disturbance in a sound wave is a series of high-pressure and low-pressure regions that travel through air
What are rarefactions?
low-pressure regions they also propagate along the tube, following the compressions
What are the highest and lowest points of a wave called?
A point at which the displacement of the element from its normal position is highest is called the crest of the wave. The lowest point is called the trough.
What is a transverse wave?
A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called transverse. Notice that no part of the string ever moves in the direction of the propagation.
What is a longitudinal wave?
A traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called longitudinal. The left end of the spring is pushed briefly to the right and then pulled briefly to the left. This movement creates a sudden compression of a region of the coils. The compressed region travels along the spring. Notice that the direction of the displacement of the coils is parallel to the direction of propagation of the compressed region. A traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called longitudinal.
What do all mechanical waves require?
All mechanical waves require (1) some source of disturbance, (2) a medium containing elements that can be disturbed, and (3) some physical mechanism through which elements of the medium can influence each other.
What does an earthquake represent?
An earthquake represents a disturbance that results in seismic waves. Two types of three-dimensional seismic waves travel out from a point under the Earth's surface at which an earthquake occurs: transverse and longitudinal. ......The longitudinal waves are the faster of the two, traveling at speeds in the range of to km/s near the surface. They are called P waves, with "P" standing for primary, because they travel faster than the transverse waves and arrive first at a seismograph (a device used to detect waves due to earthquakes). The slower transverse waves, called S waves, with "S" standing for secondary, travel through the Earth at to km/s near the surface. By recording the time interval between the arrivals of these two types of waves at a seismograph, the distance from the seismograph to the point of origin of the waves can be determined. This distance is the radius of an imaginary sphere centered on the seismograph. The origin of the waves is located somewhere on that sphere. The imaginary spheres from three or more monitoring stations located far apart from one another intersect at one region of the Earth, and this region is where the earthquake occurred.
What are the transverse speed and transverse acceleration of elements of the string?
An element at point P (or any other element of the string) moves only vertically, and so its x coordinate remains constant. Therefore, the transverse speed Vy (not to be confused with the wave speed ) and the transverse acceleration ay of elements of the string are Vy=-𝟂Acos(kx-𝟂t) ay=-𝟂^2Asin(kx-𝟂t)
What is the size of an ideal particle?
An ideal particle has zero size.
What is an ideal wave?
An ideal wave has a single frequency and is infinitely long; that is, the wave exists throughout the Universe. (A wave of finite length must necessarily have a mixture of frequencies.) ideal waves can be combined to build complex waves, just as we combined particles: the wave is a basic building block.
As the wave moves along the string, this amount of energy passes by a given point on the string during a time interval of one period of the oscillation. Therefore, the power P, or rate of energy transfer Tmw associated with the mechanical wave, is
P = Tmw/Δt = Eƛ/T = 1/2μ𝟂^2A^2ƛ/T = 1/2μ𝟂^2A^2 (ƛ/T) P= 1/2 μ𝟂^2A^2v
What speed do the low-pressure and high-pressure regions travel at?
Both regions move at the speed of sound in the medium.
When does destructive interference occur?
Destructive interference occurs when the displacements of the waves are in opposite direction, tending to cancel one another out.
At what sound level are ear plugs reccomended?
Ear plugs are recommended whenever sound levels exceed 90 dB. Recent evidence suggests that "noise pollution" may be a contributing factor to high blood pressure, anxiety, and nervousness.
What does the speed of sound depend on for longitudinal sound waves in a solid rod of material?
For longitudinal sound waves in a solid rod of material, for example, the speed of sound depends on Young's modulus Y and the density 𝛒.
What do we use a logarithmic scale for?
For the wide range of intensities oh the human ear β=10 log (I/Io) The constant Io is the reference intensity, taken to be at the threshold of hearing (Io=1.00 x 10^-12 W/m^2), and I is the intensity in watts per square meter to which the sound level β corresponds, where β is measured in decibels (dB)
The element of the string is modeled as a particle under a net force. Therefore, applying Newton's second law to this element in the radial direction gives what equation?
Fr=mv^2/R → 2Tθ=2μRθv^2/R → T=μv^2 solving for v: v=√(T/μ)
What happens if the source is moving directly toward the observer but to the right of the origin position of the previous wave?
If the source moves directly toward observer, each new wave is emitted from a position to the right of the origin of the previous wave. As a result, the wave fronts heard by the observer are closer together than they would be if the source were not moving.
What is the simple harmonic movement/motion from?
If we focus on one element of the medium, such as the element at x=0 , we see that each element moves up and down along the axis in simple harmonic motion. This movement is the motion of the elements of the medium.
What are other characteristics of a period T?
If you count the number of seconds between the arrivals of two adjacent crests at a given location in space, you measure the period T of the waves the period is the time interval required for an element of the medium to undergo a complete cycle and return to the same position The period of the wave is the same as the period of the simple harmonic oscillation of one element of the medium.
Consider two waves propagating to the right on a string. Both have the same amplitude and frequency, but they differ in phase by some amount φ y (x, t) = 2Acos φ/2 sin (kx- 𝟂t + ɸ/2)
Notice how the equation still describes a wave traveling on a string but with an amplitude that depends on the phase between the two individual waves
What is the displacement of the element along?
Notice that the displacement of the element is along x, in the direction of propagation of the sound wave.
What is one way to demonstrate wave motion?
One way to demonstrate wave motion is to flick one end of a long string that is under tension and has its opposite end fixed. In this manner, a single bump (called a pulse) is formed and travels along the string with a constant speed The hand is the source of the disturbance. The string is the medium through which the pulse travels—individual elements of the string are disturbed from their equilibrium position Furthermore, the elements of the string are connected together so they influence each other: as one element goes up, it pulls the next one upward.
How many dimensions do sound waves move through?
Sound waves can move through three-dimensional bulk media
What is the average power emitted by a spherical wave?
The average power emitted by the source must be distributed uniformly over each spherical wave front of area 4𝛑r^2, where r is the distance from the point source to the wave front. Hence, the wave intensity at a distance from the source is I=(Power)avg/A=(Power)avg/4𝛑r^2 The intensity decreases as the square of the distance from the source. This inverse-square law is reminiscent of the behavior of gravity
(a) An undisturbed cylindrical element of gas of length in a tube of cross-sectional area . (b) When a sound wave propagates through the gas, the element is moved to a new position and has a different length. The parameters and describe the displacements of the ends of the element from their equilibrium positions.
The cylinder's two flat faces move through different distances s1 and s2. the change in volume ΔV of the element in the new position is equal to AΔs, where Δs=s1-s2
What is 𝟇?
The phase constant This constant can be determined from the initial conditions.
What is the relationship between the pressure variation and the displacement in a sound wave?
The pressure variation is a maximum when the displacement from equilibrium is zero, and the displacement from equilibrium is a maximum when the pressure variation is zero.
What are some of the characteristics of the pulse? (regarding height, speed and shape)
The pulse has a definite height and a definite speed of propagation along the medium. The shape of the pulse changes very little as it travels along the string.
What is the radial distance between adjacent wave fronts that have the same phase?
The radial distance between adjacent wave fronts that have the same phase is the wavelength 𝛌 of the wave.
What are rays?
The radial lines pointing outward from the source, representing the direction of propagation of the waves, are called rays.
Does the shape of the pulse change as it travels down the string?
The shape of the pulse changes very little as it travels along the string.
Is the speed of sound affected by temperature?
The speed of sound also depends on the temperature of the medium. For sound traveling through air, the relationship between wave speed and air temperature is v= 331 √1+ (Tc/273) where v is in meters/second, 331 m/s is the speed of sound in air at 0 degrees celcius , and Tc is the air temperature in degrees Celsius. Using this equation, one finds that at 20 degrees celsius, the speed of sound in air is approximately 343 m/s.
what is the superposition principle?
The superposition principle states that the resulting disturbance in a medium is equal to the algebraic sum of the individual disturbance due to each wave at that point in time.
Imagine undergoing an infinitesimal displacement outward from the right end of the rope element along the blue line representing the force (sort of up/right) . This displacement has infinitesimal and components and can be represented by the vector dx Î + dy ĵ. What is the tangent of the angle with respect to the x axis?
The tangent of the angle with respect to the axis for this displacement is dy/dx
What is the period T of a wave?
The time interval required for the element to complete one cycle of its oscillation and for the wave to travel one wavelength
what do the transverse and longitudinal displacements in a water wave represent?
The transverse displacements seen in represent the variations in vertical position of the water elements. The longitudinal displacements represent elements of water moving back and forth in a horizontal direction
Imagine a source vibrating such that it influences the medium that is in contact with the source. Such a source creates a disturbance that propagates through the medium.If the source vibrates in simple harmonic motion with period , sinusoidal waves propagate through the medium at a speed given by:
V=ƛ/T=ƛ𝑓 ƛ=wavelength 𝑓=its frequency
What waves can be represented as linear waves?
Waves for which the amplitude A is small relative to the wavelength ƛ can be represented as linear waves.
Because we assume the shape of the wave stays constant what can we assume at different times?
We assume the shape of the pulse does not change with time. Therefore, at time t, the shape of the pulse is the same as it was at time as at time t=0
What can we conclude about a pulse or a wave of any shape?
We conclude that a pulse or a wave of any shape will travel on the string with speed v=√(T/μ), without any change in pulse shape.
How do we define intensity I of a wave?
We define the intensity I of a wave, or the power per unit area, as the rate at which the energy transported by the wave transfers through a unit area A perpendicular to the direction of travel of the wave: I= (power)avg/A I= 1/2 𝛒𝟂^2 s^2max v I = (ΔPmax)^2/2𝛒v the intensity of a periodic sound wave is proportional to the square of the displacement amplitude and to the square of the angular frequency.
Description of figure for modeling sounds waves:
We describe pictorially the motion of a one-dimensional longitudinal sound pulse moving through a long tube containing a compressible gas .A piston at the left end can be quickly moved to the right to compress the gas and create the pulse. Before the piston is moved, the gas is undisturbed and of uniform density. When the piston is pushed to the right, the gas just in front of it is compressed; the pressure and density in this region are now higher than they were before the piston moved. When the piston comes to rest, the compressed region of the gas continues to move to the right, corresponding to a longitudinal pulse traveling through the tube with speed v .
Do waves transport energy?
Yes, Waves transport energy Tmv through a medium as they propagate. For example, suppose an object is hanging on a stretched string and a pulse is sent down the string. When the pulse meets the suspended object, the object is momentarily displaced upward. In the process, energy is transferred to the object and appears as an increase in the gravitational potential energy of the object-Earth system.
What is a fun fact about ultrasonic sound waves?
You may have used a "silent" whistle to retrieve your dog. Dogs easily hear the ultrasonic sound this whistle emits, although humans cannot detect it at all. Ultrasonic waves are also used in medical imaging.
By definition, the wave travels through a displacement Δx equal to one wavelength ƛ in a time interval Δt of one period T. Therefore, the wave speed, wavelength, and period are related by the expression
v=Δx/Δt=ƛ/T
What is the general expression for the speed of all mechanical waves?
v=√(elastic property/inertial property)
What equations can we use to express the wave speed v?
v=𝟂/k v=ƛf
What does the wave speed of a linear mechanical wave soley depend on?
wave speed depends only on the properties of the medium through which the wave travels Waves for which the amplitude A is small relative to the wavelength ƛ can be represented as linear waves.
What are sound waves?
waves which travel through any material, but are most commonly experienced as the mechanical waves traveling through air that result in the human perception of hearing
What functions have we used to express the displacement and pressure?
we have expressed the displacement by means of a cosine function and the pressure by means of a sine function.
Increasing the intensity of a sound by a factor of 100 causes the sound level to increase by what amount? a. 100 dB b. 20 dB c. 10 dB d. 2 dB
b. 20 dB the factor of 100 is two powers of 10, thus the log of 100 is 2, which is multiplied by 10 which gives 20 dB
What 3 categories are sound wave that cover different frequency ranges divided into?
(1) Audible waves - lie within the range of sensitivity of the human ear. They can be generated in a variety of ways, such as by musical instruments, human voices, or loudspeakers. (2) Infrasonic waves - have frequencies below the audible range. Elephants can use infrasonic waves to communicate with one another, even when separated by many kilometers. (3) Ultrasonic waves - have frequencies above the audible range.
What is the average power over one period of oscilaation?
(Power)avg= 1/2 𝛒A𝟂^2 s^2max v A here is the area of the piston
What is the rate of energy transfer by a sinusoidal wave on a string is proportional to?
(a) the square of the frequency, (b) the square of the amplitude, and (c) the wave speed.
An 80.0-kg hiker is trapped on a mountain ledge following a storm. A helicopter rescues the hiker by hovering above him and lowering a cable to him. The mass of the cable is 8.0 kg, and its length is 15.0 m. A sling of mass 70.0 kg is attached to the end of the cable. The hiker attaches himself to the sling, and the helicopter then accelerates upward. Terrified by hanging from the cable in midair, the hiker tries to signal the pilot by sending transverse pulses up the cable. A pulse takes 0.250 s to travel the length of the cable. What is the acceleration of the helicopter? Assume the tension in the cable is uniform.
-Imagine the effect of the acceleration of the helicopter on the cable. The greater the upward acceleration, the larger the tension in the cable. In turn, the larger the tension, the higher the speed of pulses on the cable. -This problem is a combination of one involving the speed of pulses on a string and one in which the hiker and sling are modeled as a particle under a net force. v=√(T/μ)→ T=μv^2 𝛴F=ma → T-mg=ma a=T/m-g = μv^2/m -g =mcable/lcable m (Δx/Δt)^2-g a=(8.00 kg)/(15.0 m) (150.0 kg) (15.0 m/0.250s)^2 - 9.80 m/s^2 = 3.00 m/s^2
What is a compression?
-a compressed region gas is compressed and the density and pressure are above their equilibrium values. A compressed region is formed whenever the piston is pushed into the tube. This compressed region, called a compression, moves through the tube, continuously compressing the region just in front of itself.
A uniform string has a mass of 0.300 kg and a length of 6.00 m. The string passes over a pulley and supports a 2.00-kg object. How would you find the speed of a pulse traveling along this string? What if the block were swinging back and forth with respect to the vertical like a pendulum? How would that affect the wave speed on the string?
-the hanging block establishes a tension in the horizontal string. This tension determines the speed with which waves move on the string. -To find the tension in the string, we model the hanging block as a particle in equilibrium. Then we use the tension to evaluate the wave speed on the string T=mg v=√(T/μ) = √(mblock)(g)(l)/(mstring) v=19.8 m/s The swinging block is categorized as a particle under a net force. The magnitude of one of the forces on the block is the tension in the string, which determines the wave speed. As the block swings, the tension changes, so the wave speed changes. When the block is at the bottom of the swing, the string is vertical and the tension is larger than the weight of the block because the net force must be upward to provide the centripetal acceleration of the block. Therefore, the wave speed must be greater than 19.8 m/s. When the block is at its highest point at the end of a swing, it is momentarily at rest, so there is no centripetal acceleration at that instant. The block is a particle in equilibrium in the radial direction. The tension is balanced by a component of the gravitational force on the block. Therefore, the tension is smaller than the weight and the wave speed is less than m/s.
To double loudness approximately how much of an increase of sound level in dB is required?
As a rule of thumb, a doubling in loudness is approximately associated with an increase in sound level of 10 dB.
What happens as sound waves travel through air
As sound waves travel through air, elements of air are disturbed from their equilibrium positions. Accompanying these movements are changes in density and pressure of the air along the direction of wave motion. If the source of the sound waves vibrates sinusoidally, the density and pressure variations are also sinusoidal. The mathematical description of sinusoidal sound waves is very similar to that of sinusoidal waves on strings.
What is the change in volume ΔV of the element in the new position is equal to? AΔa, where Δs=s1-s2
AΔs, where Δs=s1-s2
How does the crest of the wave travel?
Because the speed of the pulse is v, the crest of the pulse has traveled to the right a distance vt at the time t
How is a spherical wave created?
Consider the special case of a point source emitting sound waves equally in all directions. If the air around the source is perfectly uniform, the sound power radiated in all directions is the same, and the speed of sound in all directions is the same. The result in this situation is called a spherical wave.
When does constructive interference occur?
Constructive interference occurs when the displacements due to each wave are in the same direction.
When are harmonics generated?
Generated when subsequent resonant frequencies are set in oscillation
What other type of energy in addition to kinetic energy is associated with the displacement of each element of the string? What is an expression of analysis for that energy? What is the total energy in a wavelength?
In addition to kinetic energy, there is potential energy associated with each element of the string due to its displacement from the equilibrium position and the restoring forces from neighboring elements. analysis for the total potential energy in one wavelength gives exactly the same result as (Kƛ=1/4 μ 𝟂^2A^2ƛ): Uƛ=1/4 μ𝟂^2A^2ƛ The total energy in one wavelength of the wave is the sum of the potential and kinetic energies: Eƛ=Uƛ+Kƛ=1/2μ𝟂^2A^2ƛ
Explain intensity
Intensity has to do with the concentration of power on or through a surface. The higher the power output of a source, the higher intensities it will produce at a given point. The smaller the area over which the power is focused, the higher the intensity.
Are the motion of the wave and the motion of the elements of the medium the same thing? How can each be modeled?
It is important to differentiate between the motion of the wave and the motion of the elements of the medium. An element of the medium is described by the particle in simple harmonic motion model. A point on the wave, such as the crest, can be described with the particle under constant velocity model.
How is a standing wave formed?
It is possible that a wave that is reflected from some barrier (interface) back onto the incoming wave could interfere constructively with that incoming wave. If it does so, a particular shape for the wave is reinforced and the wave form will begin to appear static...a so-called standing wave. Equations
What is the total kinetic energy Kƛ in one wavelength?
Kƛ=1/4 μ 𝟂^2A^2ƛ
Are v and vy the same?
N0! Do not confuse v, the speed of the wave as it propagates along the string, with vy, the transverse velocity of a point on the string. The speed v is constant for a uniform medium, whereas vy varies sinusoidally.
Does doppler effect depend on distance?
NO Although the intensity of a sound varies as the distance changes, the apparent frequency depends only on the relative speed of source and observer. As you listen to an approaching source, you will detect increasing intensity but constant frequency. As the source passes, you will hear the frequency suddenly drop to a new constant value and the intensity begin to decrease.
What do sound waves represent?
Naturally, we would expect sound waves to also represent a transfer of energy.
When are Newton's laws valid?
Newton's laws are valid in any inertial reference frame.
Give an example of the doppler effect in terms of a car's horn as it approaches and moves past you.
Perhaps you have noticed how the sound of a vehicle's horn changes as the vehicle moves past you. The frequency of the sound you hear as the vehicle approaches you is higher than the frequency you hear as it moves away from you.
How would stiffness affect wave speed?
Stiffness represents a restoring force in addition to tension and increases the wave speed.
As the piston oscillates sinusoidally, regions of compression and rarefaction are continuously set up. What does the distance between two successive compressions (or two successive rarefactions) equal?
The distance between two successive compressions (or two successive rarefactions) equals the wavelength ƛ of the sound wave.
What is the distance from one crest to the next called?
The distance from one crest to the next is called the wavelength ƛ (Greek letter lambda ƛ). More generally, the wavelength is the minimum distance between any two identical points on adjacent waves
What is the disturbance in a sound wave a series of?
The disturbance in a sound wave is a series of high-pressure and low-pressure regions that travel through air
What mass does the element have?
The element has mass m=μ𝚫s, where μ is the mass per unit length of the string. Because the element forms part of a circle and subtends an angle of 2θ at the center, Δs=R(2θ) , and m=μΔs=2μRθ
How do we model the element of gas in terms of momentum?
The element of gas is modeled as a nonisolated system in terms of momentum
What is the frequency condition to produce standing waves?
The frequency condition to produce standing waves is a frequency for which the waveform has points of no oscillation at the ends of the string
What is the frequency of the wave the same as?
The frequency of the wave is the same as the frequency of the simple harmonic oscillation of one element of the medium.
What 2 variables does the wave function depend on?
The function y(x,t), sometimes called the wave function, depends on the two variables x and t. For this reason, it is described as "y as a function of x and t."
Consider a sinusoidal wave traveling on a string. The source of the energy is the vibrating blade at the left end of the string. We can consider the string to be a non isolated system. As the blade performs work on the end of the string, moving it up and down, energy enters the system of the string and propagates along its length. Let's focus our attention on an infinitesimal element of the string of length dx and mass dm. We can model each element of the string as a particle in simple harmonic motion, with the oscillation in the y direction. All elements have the same angular frequency 𝟂 and the same amplitude A.
The kinetic energy K associated with a moving particle is K=1/2 mv^2. If we apply this equation to the infinitesimal element, the kinetic energy dK associated with the up and down motion of this element is dK=1/2 (dm)vy^2 vy=transverse speed of the element If μ is the mass per unit length of the string, the mass dm of the element of length dx is equal to μdx. Hence, we can express the kinetic energy of an element of the string as dK=1/2 (μ dx)vy^2
What are the maximum magnitudes of the transverse speed and transverse acceleration?
The maximum magnitudes of the transverse speed and transverse acceleration are simply the absolute values of the coefficients of the cosine and sine functions: a(y)(max)=𝟂^2A v(y)(max)=𝟂^2A
What is the most common unit for frequency f and period T?
The most common unit for frequency is s^(-1), or hertz (Hz). The corresponding unit for T is seconds.
What would happen to a wave if we changed its expression from y(x,t)=2/(x-3.0t)^2 +1 to y(x,t)=4/(x+3.0t) +1
The new expression represents a pulse with a similar shape, but moving to the left as time progresses. (because of the change from plus sign minus sign) Another new feature here is the numerator of 4 rather than 2. Therefore, the new expression represents a pulse with twice the height of the original function.
What can be a considered a basic building block?
The particle
Do the transverse speed and transverse acceleration of elements of the string reach their maximum values simultaneously? When do they reach their maximum values?
The transverse speed and transverse acceleration of elements of the string do not reach their maximum values simultaneously. The transverse speed reaches its maximum value (𝟂A) when y=0, whereas the magnitude of the transverse acceleration reaches its maximum value (𝟂^2A) when y=±A.
What does the y coordinate represent in the wave function y(x,t)?
The wave function y(x,t) represents the y coordinate—the transverse position—of any element located at position x at any time t.
Let us use a mechanical analysis to derive the expression for the speed of a pulse traveling on a stretched string under tension T . Consider a pulse moving to the right with a uniform speed v. In this reference frame, the pulse remains fixed and each element of the string moves to the left through the pulse shape. A short element of the string, of length Δs , forms an approximate arc of a circle of radius R. As it travels through the arc, we can model the element as a particle in circular motion.
This element has a centripetal (downward) acceleration of V^2/R, which is supplied by components of the force T whose magnitude is the tension in the string. The force T acts on each side of the element, tangent to the arc. The horizontal components of T cancel, and each vertical component Tsinϴ acts downward. Hence, the magnitude of the total radial force on the element is 2Tsinϴ . Because the element is small, ϴ is small and we can use the small-angle approximation sinϴ≈ϴ. Therefore, the magnitude of the total radial force is Fr=2Tsinϴ≈2T𝛳
All wave functions y(x, t) represent solutions of an equation called the linear wave equation.
This equation gives a complete description of the wave motion, and from it one can derive an expression for the wave speed. Furthermore, the linear wave equation is basic to many forms of wave motion.
What is the maximum position of an element of the medium relative to its equilibrium position called?
amplitude A
What is a result of the sound wave being longitudinal?
as the compressions and rarefactions travel through the tube, any small element of the gas moves with simple harmonic motion parallel to the direction of the wave.
A sinusoidal wave of frequency 𝑓 is traveling along a stretched string. The string is brought to rest, and a second traveling wave of frequency 2𝑓 is established on the string. describe the wavelength of the second wave a. twice that of the first wave b. half that of the first wave c. the same as that of the first wave d. impossible to determine
b, half that of the first wave v = f*λ λ =v/f since v remains the same so if we do f' = 2f λ' = v/2f = 1/2λ λ' =λ/2
A vibrating guitar string makes very little sound if it is not mounted on the guitar body. Why does the sound have greater intensity if the string is attached to the guitar body? a. The string vibrates with more energy. b. The energy leaves the guitar more rapidly. c. The sound power is spread over a larger area at the listener's position. d. The sound power is concentrated over a smaller area at the listener's position. e. The speed of sound is higher in the material of the guitar body. f. None of these answers is correct.
b. The energy leaves the guitar more rapidly. A guitar string vibrating by itself does not produce a very loud sound. The string itself disturbs very little air since its small surface area makes very little contact with surrounding air molecules. However, if the guitar string is attached to a larger object, such as a wooden sound box, then more air is disturbed. The sound bounces around inside the instrument and out through the hole, producing pressure difference on nearby molecules and they vibrate. Energy is same, but it propgates faster thus increasing intensity
Why do you observe a different frequency when at rest compared to when moving?
because 𝑓=1/T When you are moving the period is shorter because you interact with the wavelength faster than when when stationary, so T decreases. (say you are in a stationary boat, each wavelength hits you every 3 seconds, but you start moving toward the wave and it hits you sooner than 3 seconds, hence the period has decreased)
How can one produce a one-dimensional periodic sound wave in the tube of gas
by causing the piston to move in simple harmonic motion.
The amplitude of a wave is doubled, with no other changes made to the wave. As a result of this doubling, which of the following statements is correct? a. The speed of the wave changes. b. The frequency of the wave changes. c. The maximum transverse speed of an element of the medium changes. d. Statements (a) through (c) are all true. e. None of statements (a) through (c) is true.
c, The maximum transverse speed of an element of the medium changes.
A sinusoidal wave of frequency 𝑓 is traveling along a stretched string. The string is brought to rest, and a second traveling wave of frequency 2𝑓 is established on the string. What is the wave speed of the second wave? A. twice that of the first wave B. half that of the first wave C. the same as that of the first wave D. impossible to determine
c, the same as the first wave for speed, in a string, we use this formula v = √ T/μ m/s T = tension in string and μ = m/L mass per unit length kg/m so T and μ are both string property so in both case, they remain unchanged so speed stays the same
If you blow across the top of an empty soft-drink bottle, a pulse of sound travels down through the air in the bottle. At the moment the pulse reaches the bottom of the bottle, what is the correct description of the displacement of elements of air from their equilibrium positions and the pressure of the air at this point? a. The displacement and pressure are both at a maximum. b. The displacement and pressure are both at a minimum. c. The displacement is zero, and the pressure is a maximum. d. The displacement is zero, and the pressure is a minimum.
c. The displacement is zero, and the pressure is a maximum.
What does adjusting the frequency of oscillation do?
changes the wavelength and can give ride to different modes of oscillation.
A sinusoidal wave of frequency 𝑓 is traveling along a stretched string. The string is brought to rest, and a second traveling wave of frequency 2𝑓 is established on the string. describe the amplitude of the second wave. a. twice that of the first wave b. half that of the first wave c. the same as that of the first wave d. impossible to determine
d, impossible to determine amplitude will depend on initial conditions like how much you pull the string indirectly it depends on the energy supplied to the string so from this data it is impossible to determine
Which of the following, taken by itself, would be most effective in increasing the rate at which energy is transferred by a wave traveling along a string? a. reducing the linear mass density of the string by one-half b. doubling the wavelength of the wave c. doubling the tension in the string d. doubling the amplitude of the wave
d. doubling the amplitude of the wave Answer is (d). Doubling the amplitude of the wave causes the power to be larger by a factor of 4 In (a), halving the linear mass density of the string causes the power to change by a factor of 0.71—> the rate decreases. In (b), doubling the wavelength of the wave halves the frequency and causes the power to change by a factor of 0.25 —> the rate decreases In (c), doubling the tension in the string changes the wave speed and causes the power to change by a factor of 1.4 —> not as large as in part(d)
What is the kinetic energy of a given element if we take a snapshot of the wave at time t=0?
dK=1/2 μ𝟂^2A^2cos^2kx dx
You stand on a platform at a train station and listen to a train approaching the station at a constant velocity. While the train approaches, but before it arrives, what do you hear? a. the intensity and the frequency of the sound both increasing b. the intensity and the frequency of the sound both decreasing c. the intensity increasing and the frequency decreasing d. the intensity decreasing and the frequency increasing e. the intensity increasing and the frequency remaining the same
e. the intensity increasing and the frequency remaining the same There frequency depends only speed but not depend on distance.
Suppose you create a pulse by moving the free end of a taut string up and down once with your hand beginning at t=0. The string is attached at its other end to a distant wall. The pulse reaches the wall at time t. Which of the following actions, taken by itself, decreases the time interval required for the pulse to reach the wall? More than one choice may be correct a. moving your hand more quickly, but still only up and down once by the same amount b. moving your hand more slowly, but still only up and down once by the same amount c. moving your hand a greater distance up and down in the same amount of time d. moving your hand a lesser distance up and down in the same amount of time e. using a heavier string of the same length and under the same tension f. using a lighter string of the same length and under the same tension g. using a string of the same linear mass density but under decreased tension h. using a string of the same linear mass density but under increased tension
f and h -using a lighter string of the same length and under the same tension -using a lighter string of the same length and under the same tension
What is the inverse of the period T? What information does it give?
frequency, f The same information is more often given by the inverse of the period, which is called the frequency f . the frequency of a periodic wave is the number of crests (or troughs, or any other point on the wave) that pass a given location in a unit time interval. The frequency of a sinusoidal wave is related to the period by the expression f=1/T
What does interference describe?
how waves combine
what frequencies are humans sensitive to?
humans are sensitive to frequencies ranging from about 20 Hz to about 20,000 Hz.
How can a sound wave in a gas be described?
in terms of either pressure or displacement
What if we replaced the hand with an oscillating blade whose end is vibrating in simple harmonic motion attached to the string? What intervals does it create? How does each element move? What is the frequency like?
intervals of T/4 Because the end of the blade oscillates in simple harmonic motion, each element of the string, such as that at P, also oscillates vertically with simple harmonic motion. Therefore, every element of the string can be treated as a simple harmonic oscillator vibrating with a frequency equal to the frequency of oscillation of the blade.
what is the equation for angular wave number k? (aka the wave number)
k=2π/ƛ
what type of wave is a sinusoidal wave a simple example of?
periodic continuous wave The sinusoidal wave is the simplest example of a periodic continuous wave and can be used to build more complex waves
What is a sinusoidal wave? How is it produced? Why is it called a sinusoidal wave? What type of motion has a close relationship to sinusoidal waves?
produced by shaking the end of the rope up and down continuously in simple harmonic motion wave represented by this curve is called a sinusoidal wave because the curve is the same as that of the function sin θ plotted against θ Because shaking the end of the rope in simple harmonic motion leads to a sinusoidal wave, we see that there is a close relationship between simple harmonic motion and sinusoidal waves.
What are the modes of oscillation that give rise to standing waves called?
resonant frequencies
If s(x, t) is the position of a small element relative to its equilibrim position, what can we express this harmonic position function as?
s(x, t) = s(max) cos (kx-𝟂t) where s(max) is the maximum position of the element relative to equilibrium This parameter is often called the displacement amplitude of the wave. The parameter k is the wave number, and 𝟂 is the angular frequency of the wave
What happens when the source moves away from a stationary observer?
the observer measures a wavelength λ' that is greater than λ and hears a decreased frequency: 𝑓 ' = (v/v+vs) 𝑓
How does the speed of sound in a gas relate to the speed of transverse waves on a string?
speed of transverse waves on a string: v= √(T/𝛍) In both cases, the wave speed depends on an elastic property of the medium (bulk modulus B or string tension T) and on an inertial property of the medium (volume density 𝛒 or linear density 𝛍). In fact, the speed of all mechanical waves follows an expression of the general form v=√(elastic property/inertial property)
What happens if a point source emits sounds waves and the medium is uniform, the waves move at the same speed in all directions radially away from the source? what is the distance between adjacent wave fronts equal to?
spherical wave The distance between adjacent wave fronts equals the wavelength 𝛌
What does the wave speed of linear mechanical waves solely depend on?
that the wave speed depends only on the properties of the medium through which the wave travels
What is a wave front?
the arcs may connect corresponding crests on all the waves. We call such a surface of constant phase a wave front.
What is the essence of wave motion?
the essence of wave motion: the transfer of energy through space without the accompanying transfer of matter.
What happens when the piston is pulled back?
the gas in front of it expands and the pressure and density in this region fall below their equilibrium values
What is the fundamental frequency?
the lowest frequency to meet the frequency condition of standing waves: the frequency condition to produce standing waves is a frequency for which the waveform has points of no oscillation at the ends of the string
what is the expression for the speed of sound in a gas?
v= √(B/𝛒)
How out of phase is the pressure wave with the displacement wave?
the pressure wave is 90 degrees out of phase with the displacement wave The pressure variation is a maximum when the displacement from equilibrium is zero, and the displacement from equilibrium is a maximum when the pressure variation is zero.
sound waves in gases the scenario: consider a cylindrical element of gas between the piston and the dashed line. This element of gas is in equilibrium under the influence of forces of equal magnitude, from the piston on the left and from the rest of the gas on the right. The magnitude of each of these forces is PA, where P is the pressure in the gas and A is the cross-sectional area of the tube. The length of the undisturbed element of gas is chosen to be vΔt, where is the speed of sound in the gas and Δt is the time interval between the configurations
the situation after this time interval Δt, during which the piston moves to the right at a constant speed Vx due to a force from the left on the piston that has increased in magnitude to (P+ΔP)A. Because the speed of sound is v, the sound wave will just reach the right end of the cylindrical element of gas at the end of the time interval Δt. The gas to the right of the element is undisturbed because the sound wave has not reached it yet. At this moment every bit of gas in the element is moving with speed Vx.
what is the threshold intensity, and what intensity level does it correspond to?
the threshold intensity is 10^-12 W/m^2, corresponding to an intensity level of 0 dB.
What happens to the observed λ and actual λ If the source is moving directly toward the observer but to the right of the origin position of the previous wave?
the wavelength λ' measured by observer A is shorter than the wavelength λ of the source. During each vibration, which lasts for a time interval T (the period), the source moves a distance vsT=vs/f and the wavelength is shortened by this amount. Therefore, the observed wavelength λ' is λ'=λ-Δλ=λ-vs/𝑓 𝑓 ' = (v/v-vs) 𝑓 That is, the observed frequency is increased whenever the source is moving toward the observer.
What happens when 2 waves attempt to occupy the same space?
they combine into one wave whose shape is governed by the superposition principle
Consider "the wave" at a baseball game: people stand up and raise their arms as the pulse arrives at their location, and the resultant pulse moves around the stadium. Is this pulse a. transverse or b. longitudinal
transverse
How can a sinusodial wave be expressed?
y(x,t)=Asin(kx-𝟂t) A=amplitude k=wave number 𝟂=angular frequency
The wave function given by the equation y(x,t)= Asin(kx-𝟂t) assumes the vertical position y of an element of the medium is zero at x=0 at and t=0. That need not be the case. If it is not, we generally express the wave function in the form:
y(x,t)=Asin(kx-𝟂t+𝟇)
What expression can we use to describe the motion of any element of the string?
y=Asin(kx-𝟂t) y=y(x,t)
Is frequency higher when you are moving closer to or further from the wave according to the doppler effect?
you observe a lower frequency than when at rest when moving away from the wave you observe a higher frequency than when at rest when moving toward the wave because you are moving away, it takes a longer period T to interact with the wave, and you observe a lower frequency
Using the bulk modulus, how can we express the pressure variation in the element of gas as a function of its volume?
ΔP = -B Δv/Vi what if we substitute for the initial volume and the change in volume of the element: ΔP = -B (AΔs/AΔx)
What expression shows us that a displacement described by a cosine function leads to a pressure described by a sine function.
ΔP = B smax ksin(kx-𝟂t) We also see that the displacement and pressure amplitudes are related by ΔPmax=B smax k This relationship depends on the bulk modulus of the gas, which is not as readily available as is the density of the gas.
The variation in the gas pressure ΔP measured from the equilibrium value is also periodic with the same wave number and angular frequency as for the displacement. Therefore, we can write:
ΔP = ΔPmax sin (kx-𝟂t) where the pressure amplitude ΔPmax is the maximum change in pressure from the equilibrium value.
The pressure change can be related to the volume change and then to the speeds and through the bulk modulus:
ΔP= -B(ΔV/Vi) = -B (-VxAΔt)/vAΔt = BVx/V Therefore, the impulse becomes → I = (AB (Vx/v) Δt) Î
What expression relates the relationship between pressure amplitude and displacement amplitude for a sound wave?
ΔPmax=Bs(max)k = (𝛒v^2)s(max) (𝟂/v)
On the right of the impulse momentum therom, the impulse is provided by the constant force due to the increased pressure on the piston:
→. → I = 𝛴 F Δt = (AΔPΔt) Î
Consider the element of gas acted on by the piston. Imagine that the piston is moving back and forth in simple harmonic motion at angular frequency 𝟂. Imagine also that the length of the element becomes very small so that the entire element moves with the same velocity as the piston. Then we can model the element as a particle on which the piston is doing work. The rate at which the piston is doing work on the element at any instant of time is given by:
→. → Power = F * Vx → The force F on the element of gas is related to the pressure → and the velocity Vx of the element is the derivative of the displacement function, so we find Power = 𝛒v𝟂^2 As^2max sin^2 (kx-𝟂t)
The force from the piston has provided an impulse to the element, which in turn exhibits a change in momentum. Therefore, we evaluate both sides of the impulse-momentum theore:
→. → ΔP = I The change in the momentum of a particle is equal to the impulse of the net force acting on the particle
On the left-hand side of the impulse-momentum theorem, the change in momentum of the element of gas of mass is as follows:
→. → Δp = mΔv = (𝛒Vi) (VxÎ-0) = 𝛒VVxAΔt)Î we find: 𝛒VVxAΔt=AB (Vx/V)Δt which reduced to an expression for the speed of sound in a gas: v= √(B/𝛒)
What equation gives the general relationship for the observed frequency that includes all four conditions?
𝑓 ' = ( v+ vo / v - vs) 𝑓 In this expression, the signs for the values substituted for vo and vs depend on the direction of the velocity. A positive value is used for motion of the observer or the source toward the other (associated with an increase in observed frequency), and a negative value is used for motion of one away from the other (associated with a decrease in observed frequency).
Suppose a traveling wave is propagating along a string that is under a tension T. Let's consider one small string element of length 𝚫x. The ends of the element make small angles 𝛳a and 𝛳b with the x axis. Forces act on the string at its ends where it connects to neighboring elements. Therefore, the element is modeled as a particle under a net force. The net force acting on the element in the vertical direction is
𝛴Fy= Tsin𝛳b-Tsin𝛳a =T (sin𝛳b-sin𝛳a) Because the angles are small, we can use the approximation sin𝛳≈tan𝛳 to express the net force as: 𝛴Fy ≈ T(tan𝛳b-tan𝛳a)
What is the equation for the angular frequency 𝟂?
𝟂=2π/T=2πf
