Waves

Question 1

Traveling Wave

Answer 1

• Given rope, moving hand up and down will create a wave traveling along the rope called a mechanical wave (disturbance transmitted by a medium from one point to another, without the medium itself being transported)
  
  • Observer near the system would see peaks and valleys moving along the rope, in what is called a traveling wave
    
    • At any point (x) along the rope, the rope has a varying vertical displacement (y) at time t
      
      • Variation in vertical displacement defines shape of the wave
      • For a traveling wave, the displacement y of each point depends not only on x but also on t. An equation that gives y therefore must be a function of both position (x) and time (t). As y depends on two independent variables, wave analysis must be looked at from two point of views with only one variable changing
        
        • Point of View #1: x varies, t does not
        • Point of View #2: y varies,, x does not

Question 2

Point of View #1: x Varies, t Does Not

Answer 2

• To keep t from varying, we must freeze time by imagining a photograph of the wave. Taking snapshots at times t₁ and t₂, all x’s along the rope are visible (that is, x varies), but t does not.
• From this point of view, all the points at which the rope has its maximum vertical displacement (above the horizontal) called crests and all the points at which the rope has its maximum vertical displacement (below the horizontal) are called troughs.
  
  • These crests and troughs repeat themselves at regular intervals along the rope and the distance between two adjacent crests (or two adjacent troughs) is the length of one wave, and is called the wavelength, λ
• The maximum displacement from the horizontal equilibrium position of the rope is known as the amplitude, A of the wave.
  
  • A is just the distance from the “middle” to a crest, NOT from a trough to a crest

Question 3

Point of View #2: t Varies, x Does Not

Answer 3

• One position is designated x along the rope to watch as time varies. This can be done by visualizing two screens in front of the rope, with only a narrow gap between them. Obersvations can be made on how a single point on the rope varies as a wave travels behind the screen. (wave is traveling because time is not frozen). This point on the rope moves up and down. Since the direction in which the rope oscillates (vertically) is perpendicular to the direction in which the wave propagates (travels horrizontally), this wave is tranverse.
  
  • ​The time it takes for one complete vertical oscillation of a point on the rope is called the period, T, of the wave, and the number of cycles it completes in one second is called its frequency, f. The period and frequency are established by the source of the wave and T= 1/f

Question 4

Wave Speed

Answer 4

• Five important characteristics of any wave are its wavelength, amplitude, peroid, frequency, and speed, v
• In this case of point of view #2, the time it takes for the visible point on the rope to move from its crest position, down to its trough position, and then back up to the crest position is its period, T. Behind the scree, the wave moved a distance of one wavelength. T is the time required for one wave to travel by a point, and λ is the distance traveled by one wave.
  
  • Therefore the equation distance= rate x time becomes:
    
    • λ= vT
    • λ•1/T= v
      
      • λf= v
      • The simple equation v= λf shows how the wave speed, wavelength, and frequency are interconnected (frequency, period, wavelength, and wave speed have nothing to do with amplitude). It’s the most basic equation in wave theory

Question 5

Medium

Answer 5

• Do not be fooled by formula, the speed of a wave does not depend on the frequency or wavelength, it depends on properties of the medium.
  
  • Wave Rule #1: All waves of the same type in the same medium have the same speed.
• If you move a rope up and down at a certain frequency and then suddenly increase that frequency, the speed remains unchanged. The wavelength, therefore, decreases.
  
  • More pulses per second are created, each pulse won’t have as much time to move down the rope before the next pulse is created. The pulses are closer together, meaning the wavelength has decreased

Question 6

Wave Speed on a Stretched Spring

Answer 6

• An equation can be derived for the speed of a transverse wave on a stretched string or rope. The mass of the string m and its length L; its linear mass density (μ) is m/L
  
  • If the tension in the string is F_T, then the speed of a travelling tranverse wave on this string is given by:
    
    • v= √(F_T/μ)
      
      • v depends only on the physical characteristics of the string— its tension and linear density. So, because v= λf for a given stretched string, varying f will create different waves that have different wavelenghts, but v will not vary

Question 7

Wavefronts

Answer 7

• Wave Rule #2: When a wave passes into a new medium, its frequency stays the same.
• Waves can be represented by a series of wavefronts— lines that represent, say, the crests.
  
  • If a wave moves into a new medium, its speed will most likely change. However, the number of wavefronts that approach the boundary per second must be equal to the number of wavefronts that leave the boundary (that is, rate in equals rate out). The number of wave fronts per second is the frequency.
    
    • While the number of wavefronts leaving the first medium per second equals the number entering the second medium per second, their spacing (wavelength) changes

Question 8

Superposition of Waves

Answer 8

• When two or more waves meet, the displacement at any point of the medium is equal to the sum of the displacements due to the individual waves, this is known as superposition. When waves meet and overlap (interfere), the displacement of the string is equal to the sum of the individual displacements, but after they pass, the wave pulses continue, unchanged.
• In general, waves will be somewhere in between exactly in phase and exactly out of phase

Question 9

Constructive Interference

Answer 9

• If the two waves have displacements of the same sign when they overlap, the combined wave will have a displacement of greater magnitude than either individual wave, this is called constructive interference.
  
  • If the waves are exactly in phase— that is, if the crest meets crest and trough meets trough— then the waves will constructively interfere completely, and the amplitude of the combined wave will be the sum of the individual amplitudes

Question 10

Destructive Interference

Answer 10

• If the waves have opposite displacements when they meet, the combined waveform will have a displacement of smaller magnitude than either individual wave, this is called destructive interference
  
  • If the waves are exactly out of phase— that is, if crest meets trough and trough meets crest— then they will destructively interfere completely, and the amplitude of the combined wave will be the difference between the individual amplitudes.

Question 11

Standing Waves

Answer 11

• When a traveling wave on a string strikes the wall, the wave will reflect and travel back in the reverse direction.
  
  • The string now supports two traveling waves; the wave that is generated at one end, which travels toward the wall, and the reflected wave.
    
    • What is actually seen is the superposition of these two oppositely directed traveling waves, which have the same frequency, amplitude, and wavelength. If the length of the string is just right, the resulting pattern will oscillate vertically and remain fixed. The crests and troughs no longer travel down the length of the string. This is a standing wave

Question 12

Nodes

Answer 12

• One end of the string is fixed to the wall, and the other end is oscillated to a negligibly small amplitude so that both ends can be considered fixed ( no vertical oscillation).
• The interference of the two traveling waves results in complete destructive interference at some points (nodes) and complete constructive interfernce at other points (antinodes). Other points have amplitudes between these extremes.
  
  • Nodes can be remembered as as areas of “no displacement” and antinodes as the opposite of that
• A difference between a traveling wave and a standing wave: every point on the string had the same amplitude as the traveling wave went by, each point on a string supporting a standing wave has an individual amplitude

Question 13

Resonant Wavelengths

Answer 13

• Nodes and antinodes always alternate, they’re equally spaced, and the distance between two succesive nodes (or antinodes) is equal to 1/2λ.
• There are three simplest standing waves that the string can support.
  
  • The first standing wave has one antinode, the second has two, and the third has three. The length of all strings being L.
    
    • For the first standing wave, L is equal to 1(1/2)λ
    • For the second standing wave, L is equal to 2(1/2)λ
    • For the third standing wave, L is equal to 3(1/2)λ
      
      • A standing wave can only form when the length of the string is a multiple of 1/2λ
        
        • L= n(1/2)λ Solving for the wavelength:
          
          • λ_n= 2L/n
          • These are called the harmonic (or resonant) wavelengths, and the integer n is known as the harmonic number

Question 14

Resonant Frequencies

Answer 14

• As frequencies can be controlled of the waves one may create, it’s more helpful to figure out the frequencies that generate a standing wave
  
  • Because λf= v, and because v is fixed by the physical characteristics of the string, the special λ’s correspond to equally special frequencies. From f_n= v/λ_n we get:
    
    • f_n= nv/(2L)
      
      • These are called the harmonic (or resonant) frequencies. A standing wave will form on a string if we create a traveling wave whose frequency is the same as a resonant frequency. The first standing wave,the one for which the harmonic number, n, is 1, is called the fundamental standing wave. The nth harmonic frequency is simply n times the fundamental frequency:
        
        • f_n= nf₁
        • Likewise the nth harmonic wavelength is equal to λ₁ divided by n. Therefore, if the fundamental frequency (or wavelength) is known, all the other resonant frequencies and wavelengths can be determined.

Question 15

Sound Waves

Answer 15

• Sound Waves are produced by the vibration of an object, such as vocal cords plucked strings, or a jackhammer
  
  • The vibrations cause pressure variations in the conducting medium (wich can be gas, liquid, or solid), and if the frequency is between 20 Hz and 20,000 Hz, the vibrations may be detected by human ears.
  • The variations in the conducting medium can be positions at which the molecules of the medium are bunched together (where the pressure is above normal), which are called compressions, and positions where the pressure is below normal, called rarefractions.

Question 16

Sound Waves vs. String Waves

Answer 16

• The molecules of the medium transmitting a sound wave move parallel to the direction of wave propagation, rather than perpendicular to it. For this reason, sound waves are said to be longitudinal.
  
  • Despite this difference, all of the basic characteristics of a wave—amplitude, wavelength, period, frequency— apply to sound waves as they did for waves on a string. λf= v also holds true.
• Because it’s difficult to draw a longitudinal wave, the pressure as a function of position is instead graphed

Question 17

Bulk Modulus

Answer 17

• The speed of a sound wave depends on the medium through which it travels, in particular, it depends on the density (p) and on the bulk modulus (B), a measure of the medium’s response to compression.
  
  • A medium that is easily compressible, like a gas, has a low bulk modulus; liquids and solids, which are much less easily compressed have signfinifcantly greater bulk modulus values.
    
    • For this reason, sound generally travels faster through solids than through liquids and faster through liquids than through gases.
      
      • v= √(B/p)

Question 18

Speed of Sound

Answer 18

• The speed of sound through air can also be written in term of air’s mean pressure, which depends on its temperature.
  
  • At room temperature (approximately 20 C) and normal atmospheric pressure, sound travels at 343 m/s. This value increases as air warms or pressure increases

Question 19

Intensity

Answer 19

• How loud sound is perceived depends on both frequency and amplitude. Given a fixed frequency, loudness is measured by intensity
  
  • Intensity: I= P/A
    
    • P is the power produced by the source and A is the area over which the power is spread.

Question 20

Decibel Level

Answer 20

• Given a point source emitting a sound wave in all directions (spherical). At a distance r, the A= 4πr² (the surface area of the sphere). Therefore I∝ 1/r²
  
  • If the listener doubles the distance to the source, the sound will be heard one-fourth as loud.
  • An alternate way of measuring loudness is with the decibel level (sometimes referred to as relative intensity)
    
    • Decibel Level: β= 10log(I/I₀)
      
      • I₀ is the threshold of hearing and is equal to 10^-12 watts. while β is measured in decibels (dB), it is dimensionless.
        
        • If a sound increases by 10 dB, the intensity increases by a factor of 10

Question 21

Beats

Answer 21

• If two waves whose frequencies are close but not identical interfere, the resulting sound modulates in amplitude, becoming loud, then soft, then loud, then soft.
  
  • This is due to the fact that as the the individual waves travel, they are in phase, then out of phase, then in phase again, and so on. By superposition, the waves interfere constructively, the amplitude increases, and the sound is loud; when the waves interfere destructively, the amplitude decreases, and the sound is soft.
    
    • Each time the waves interfere constructively, producing an increase in sound level, we say that a beat has occured. The number of beats per second, known as the beat frequency, is equal to the difference between the frequencies of the two combining sound waves
      
      • f_beat= |f₁ - f₂|
        
        • If frequencies f₁ and f₂ match, then the combined waveform doesn’t waver in amplitude, and no beats are heard. When tuning a piano, if in tune no beats will be heard, if out of tune beats will be heard.

Question 22

Resonance for Sound Waves

Answer 22

• Just as standing waves can be set up on a vibrating string, standing sound waves can be established within an enclosure. A vibrating source at one end of an air-fille tube produces sound waves that travel the length of the tube. These waves reflect off the far end, and the superposition of the forward reflected waves can produce a standing wave pattern if the length of the tube and the frequency of the waves are related in a certain way.
  
  • Air molecules at the far end of the tube can’t oscillate horizontally because they’re against a wall. The far end of the tube is a displacement node. But the other end of the tube (where the vibrating source is located) is a displacement anitnode.

Question 23

Closed End Tube

Answer 23

• The distance between an antinode and an adjacent node is always 1/4 of the wavelength.
• Standing waves can be established in a tube that’s closed at one end if the tube’s length is equal to the odd multiple of 1/4λ. The resonant wavelengths and frequencies are given by the equations:
  
  • λ_n= 4L/n for any odd integer n
  • f_n= nv/(4L) for any odd integer n
• If the far end of the tube is not sealed, standing waves can be established in the tube, because sound waes can be reflected from the open ai. A closed end is a displacement node. In this case, then, the standing wave will have two displacement antinodes (at the ends of the tube), and the resonant frequencies will be given by:
  • λ_n= 2L/n for any integer n
  • f_n= nv/(2L) for any integer n
• An open-ended tube can support any harmonic, a closed-end tube can only support odd harmonics

Question 24

The Doppler Effect

Answer 24

• When a source of sound waves and a detector are not in relative motion, the frequency that the source emits matches the frequency that the detector receives
  
  • However, if there is relative motion between the source and the detector, then the waves that the detector receives are different in frequency. If the detector moves toward the source, then the detector intercepts the waves at a rate higher than the one at which they were emitted; the detector hears a higher frequency than the source emitted; if the source moves toward the detector, the wavefronts pile up, and this results in the detector receiving waves with shorter wavelengths and higher frequencies.
• When the detector moves and not the source, there is no change in wavelength. Instead, there is a change to the speed with which the detector receives wavefronts. Conversely, if the detector is moving away from the source or if the source is moving away from the detector, than the deteted waves have a lower frequency than they hgad when they were emitted by the source.
  
  • The shift in frequency that occurs is known as the Doppler Effect.

Question 25

Doppler Effect Equations

Answer 25

F_D= (v (plus or minus) v_D)/(v (minus or plus) v_s)•f_s