Wave - II

These are material waves. 1. Sound Waves Waves on Strings, etc.: Transverse Waves Sound Waves: ANY Longitudinal Waves These are material waves.

Wave Function s(x,t) = smcos(kx-wt) y(x,t) = ymsin(kx-wt) Wave Function Transverse wave s(x,t) = smcos(kx-wt) s: The displacement from the equilibrium position The sin and cos functions are identical for the wave function, differing only in a phase constant. We use cos in this chapter. sin(q+90˚)=cosq

Pressure Amplitude ∆p(x,t) = ∆pmsin(kx-wt) ∆p: the pressure change in the medium due to compression (∆p >0) or expansion (∆p <0) ∆p(x,t) and s(x,t) are 90˚ out of phase

2. Wave Speed Transverse Waves (String): Bulk modulus 2. Wave Speed Transverse Waves (String): Tension Linear density elastic inertial Sound Waves (Longitudinal Waves): elastic inertial Bulk modulus Volume density

Bulk Modulus

3. Intensity Transverse Waves (String): Sound Waves (Longitudinal Waves): A: area intercepting the sound

Wavefront, Ray, and Spherical Waves Wavefront: Equal phase surfaces Spherical: spherical waves Planar: planar waves Ray: The line  wavefront, that indicates the direction of travel of the wavefront At large radius (far from a point source): spherical wavefront  planar wavefront

Sound Intensity for a Point Source Wavefront area at distance r from the source: A = 4pr2

The Decibel Scale The sound level b is defined as: decibel 10-12 W/m2, human hearing threshold

4. Interference For two waves from two different point sources, their phase difference at any given point depends of their PATH LENGTH DIFFERENCE ∆L x  x+l kx  kx+2p f = 0: constructive f = p: destructive other: intermediate

Constructive: m=0,1,2, ... Destructive: f = m(2p), m=0,1,2, ... f = 0: constructive f = p: destructive other: intermediate f = (m+1/2)(2p), m=0,1,2, ...

Standing Waves in a Tube BOUNDARY CONDITIONS: Closed End: s = 0, a node for s ∆p = ∆pm, an antinode for ∆p Open End: s = sm, an antinode for s ∆p = 0, a node for ∆p

The time interval between the two sounds: HRW 9P (5th ed.). A man strikes a long aluminum rod at one end. A woman at the other end with her ear close to to the rod, hears the sound of the blow twice (once through air and once through the rod), with a 0.120 s interval between. How long is the rod? Let the length of the rod be l, the speed of sound in air be v1, and the speed of sound in the rod be v2. The time interval between the two sounds: Solve for l:

∆p(x,t) = ∆p msin(kx-wt) HRW 18P (5th ed.). The pressure in a traveling sound wave is given by the equation ∆p = (1.5 Pa) sin p[(1.00 m-1)x - (330 s-1)t]. Find (a) the pressure amplitude, (b) the frequency, (c) the wavelength, and (d) the speed of the wave. s(x,t) = smcos(kx-wt) ∆p(x,t) = ∆p msin(kx-wt) (a) ∆pm = 1.5 Pa (b) f = w/2p =(330 s-1)/2 =165 Hz (c) l=2p/k = 2p /(1.00 m-1) p=2 m (d) v = lf =330 m/s

HRW 23P (5th ed.). Two point sources of sound waves of identical wavelength l and amplitude are separated by distance D = 2.0l. The sources are in phase. (a) How many points of maximum signal lie along a large circle around the sources? (b) How many points of minimum signal? The phase difference at point P: (a) Maximum: ∆f=2mp sinq = m/2 (m=0, ±1, ±2, …) Eight: 0˚, 30˚, 90˚, 150˚, 180˚, 210˚, 270˚, 330˚ (b) Eight, in between the maximums.