Wave II 1 Sound Waves on Strings etc

Wave - II

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

y(x, t) = ymsin(kx-wt) Transverse wave Wave Function 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

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

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 = 4 pr 2

The Decibel Scale The sound level b is defined as: decibel 10 -12 W/m 2, 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 f = 0: constructive f = p: destructive other: intermediate kx kx+2 p

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

HRW 9 P (5 th 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 v 1, and the speed of sound in the rod be v 2. The time interval between the two sounds: Solve for l:

HRW 18 P (5 th 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. ∆p(x, t) = ∆p msin(kx-wt) s(x, t) = smcos(kx-wt) (a) ∆pm = 1. 5 Pa (b) f = w/2 p =(330 s-1)/2 =165 Hz (c) l=2 p/k = 2 p /(1. 00 m-1) p=2 m (d) v = lf =330 m/s

HRW 23 P (5 th ed. ). Two point sources of sound waves of identical wavelength l and amplitude are separated by distance D = 2. 0 l. 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=2 mp sinq = m/2 (m=0, ± 1, ± 2, …) Eight: 0˚, 30˚, 90˚, 150˚, 180˚, 210˚, 270˚, 330˚ (b) Eight, in between the maximums.