PHY2048, Lecture 23 Wave Motion Chapter 14 Tuesday November 16th Stephen Hill (covering for Prof. Wiedenhover) Review of wave solutions The wave equation Wave superposition and interference Spatial interference and standing waves Temporal beating Sources of musical sound Doppler effect (if time) Fifth and final mini exam on Thu. (Chs. 13-15) Make up labs Nov. 29 at 12:30pm and 3:30pm in UPL107 Final exam Wed. Dec. 8th, 10am to noon, in MOR 104 Displacement }
Review - wavelength and Amplitude frequency Phase Transverse wave y ( x, t ) A cos( kx t ) angular wavenumber angular frequency 2 k 2 T Phase shift
k is the angular wavenumber is the angular frequency. 1 frequency f T 2 velocity v f k T Traveling waves on a stretched ve curve string a FT
+ve curve Fnet a Zero curve FT a +ve curve is the string's linear density, or mass per unit length. Tension FT provides the restoring force (kg.m.s-2) in the string. Tension Without tension, the wave could not propagate. The mass per unit length (kg.m-1) determines the response of the string to the restoring force (tension), through Newtorn's
2nd law. Look for combinations of FT and F that give dimensions of Look T -1 v speed (m.s ). Traveling waves on a stretched ve curve string l a FT Fnet
a +ve curve Zero curve FT a +ve curve is the string's linear density, or mass per unit length. transverse The Wave Equation mass acceleration 2 2
y y Fnet FT 2 l l 2 ma y x t Dimensionless 2 2 parameter y y proportional to FT 2 2 x t curvature
The wave equation 2 2 FT y y 2 2 x t General solution: y ( x, t ) ym sin kx t 2 y 2 k y x, t 2 x
FT 2 k 2 or or y ( x, t ) ym f kx t v v 2 y 2 y x, t 2
t 2 FT 2 v 2 k FT v The principle of superposition for waves It often happens that waves travel simultaneously through the same region, e.g.
Radio waves from many broadcasters Sound waves from many musical instruments Different colored light from many locations from your TV Nature is such that all of these waves can exist without altering each others' motion Their effects simply add This is a result of the principle of superposition, which applies to all harmonic waves, i.e., waves that obey the linear wave equation 2 2 y y v 2 2 x t 2
And have solutions: y ( x, t ) ym f kx t or ym sin kx t The principle of superposition for wavesalong the same stretched If two waves travel simultaneously string, the resultant displacement y' of the string is simply given by the summation y ' x, t y1 x, t y2 x, t where y1 and y2 would have been the displacements had the waves traveled alone. This is the principle of superposition.
Overlapping waves algebraically add to produce a resultant wave (or net wave). Overlapping waves do not in any way alter the travel of each other Interference of waves Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such that The waves will add. y1 ym sin kx t y2 ym sin kx t Interference of waves Noise canceling headphones
Interference of waves Suppose two sinusoidal waves with the same frequency and amplitude travel in the same direction along a string, such that y1 ym sin kx t y2 ym sin kx t The waves will add. If they are in phase (i.e. = 0), they combine to double the displacement of either wave acting alone. If they are out of phase (i.e. = ), they combine to cancel everywhere, since sin() = sin(). This phenomenon is called interference. Interference of waves Mathematical proof: y1 ym sin kx t y2 ym sin kx t Then:
y ' x, t y1 x, t y2 x, t ym sin kx t ym sin kx t But: So: sin sin 2sin 12 cos 12 y ' x, t 2 ym cos 12 sin kx t 12 Amplitude Wave part Phase shift Interference of waves y ' x, t 2 ym cos 12 sin kx t 12 If two sinusoidal waves of the same amplitude and
frequency travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in the same direction. If = 0, the waves interfere constructively, cos = 1 and the wave amplitude is 2ym. If = , the waves interfere destructively, cos(/2) = 0 and the wave amplitude is 0, i.e. no wave at all. All other cases are intermediate between an amplitude of 0 and 2ym. Note that the phase of the resultant wave also depends on the phase difference. Adding waves as vectors (phasors) described by amplitude and phase Interference - Standing Waves If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave.
y ' x, t y1 x, t y2 x, t ym sin kx t ym sin kx t 2 ym sin kx 12 cos t 12 x dependence t dependence This is clearly not a traveling wave, because it does not have the form f(kx t). In fact, it is a stationary wave, with a sinusoidal varying amplitude 2ymcos(t). Link Reflections at a boundary Waves reflect from boundaries. This is the reason for echoes - you hear sound reflecting back to you. However, the nature of the reflection depends on the boundary condition. For the two examples on the left, the nature of the reflection depends
on whether the end of the string is fixed or loose. Standing waves and resonance At ordinary frequencies, waves travel backwards and forwards along the string. Each new reflected wave has a new phase. The interference is basically a mess, and no significant oscillations build up. Standing waves and resonance However, at certain special frequencies, the interference produces strong standing wave patterns.
Such a standing wave is said to be produced at resonance. These certain frequencies are called resonant frequencies. Standing waves and resonance Standing waves occur whenever the phase of the wave returning to the oscillating end of the string is precisely in phase with the forced oscillations. determined by geometry Thus, the trip along the string and back should be equal to an integral number of wavelengths, i.e. 2L 2 L n or n
for n 1,2,3... v v f n , for n 1,2,3... 2L Each of the frequencies f1, f1, f1, Each etc, are called harmonics, or a harmonic series; n is the harmonic number. Standing waves and resonance Here is an example of a two-dimensional vibrating diaphragm. The dark powder shows the positions of the nodes in the vibration.
Standing waves in air columns Simplest case: - 2 open ends - Antinode at each end - 1 node in the middle 1 2 L 2 L /1 Although the wave is longitudinal, we can represent it schematically by the solid and dashed green curves. Standing waves in air columns A harmonic series 1 2 L 2 L /1 2 3
4 2L , for n 1,2,3,.... n v v f n , 2L for n 1,2,3,... Standing waves in air columns A different harmonic series 1 3
5 7 4L , for n 1,3,5,.... n v v f n , 4L for n 1,3,5,... Musical instruments Flute
Oboe Saxophone Link Musical instruments v f n 4L n = even if B.C. same at both ends of pipe/string n = odd if B.C. different at the two ends Wave interference - spatial Interference - temporal (or beats) s ( x, t ) sm cos(kx t ) In order to obtain a spatial interference pattern, we placed
two sources at different locations, i.e. we varied the first term in the phase of the waves. We can do the same in the time domain whereby, instead of placing sources at different locations, we give them different angular frequencies and . For simplicity, we analyze the sound at x = 0. s s1 s2 sm cos 1t cos 2t cos cos 2cos 12 cos 12 s 2 sm cos 12 1 2 t cos 12 1 2 t 2 sm cos ' t cos t ' 12 1 2 12 1 2 Interference - temporal (or beats) s 2 sm cos ' t cos t A maximum amplitude occurs whenever 't has the value or .
This happens twice in each time period of the cosine function. Therefore, the beat frequency is twice the frequency ', i.e. beat 2 ' 1 2 f beat 2 f ' f1 f 2 Link Doppler effect Consider a source of sound at the proper frequency, f ', moving relative to a stationary observer. The observer will hear the sound with an apparent frequency, f, which is shifted from the proper frequency according to the following equation: u f 1 f ' v Here, v is the sound velocity (~330 m/s in air), and u is the
relative speed between the source and detector. When the motion of the detector or source is towards the other, use the plus (+) sign so that the formula gives an upward shift in frequency. When the motion of the detector or source is away from the other, use the minus () sign so that the formula gives a downward shift. Mach cone angle: v sin vS Energy in traveling waves y ( x, t ) ym sin kx t Kinetic energy: vy
dK = 1/2 dm vy2 y ym cos kx t t 2 dK 12 dx ym cos 2 (kx t ) Divide both sides by dt, where dx/dt = vx dm = dx Similar expression for elastic potential energy dK 1 2 v x 2 ym2 cos2 ( kx t ) dt dU 1 2 v x 2 ym2 cos2 (kx t )
dt Pavg 2 12 v 2 ym2 cos 2 (kx t ) 2 12 v 2 ym2 12 12 v 2 ym2 Energy is pumped in an oscillatory fashion down the string Note: I dropped the subscript on v since it represents the wave speed
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