1. A wave traveling along the x axis is described mathematically by the equation

Question

1. A wave traveling along the x axis is described mathematically by the equation y = 0.17sin(2.7t + 0.55x), where y is the displacement (in meters), t is in seconds, and x is in meters. What is the speed of the wave?

2. A hunter is standing on flat ground between two vertical cliffs that are directly opposite one another. He is closer to one cliff than the other. He fires a gun and, after a while, hears three echoes. The second echo arrives 1.88 s after the first, and the third echo arrives 1.69 s after the second. Assuming that the speed of sound is 343 m/s and that there are no reflections of sound from the ground, find the distance (in m) between the cliffs.

3. The security alarm on a parked car goes off and produces a frequency of 1170 Hz. The speed of sound is 343 m/s. As you drive toward this parked car, pass it, and drive away, you observe the frequency to change by 113 Hz. At what speed are you driving?

4.You are flying in an ultra-light aircraft at a speed of 49.9 m/s. An eagle, whose speed is 18.7 m/s, is flying directly toward you. Each of the given speeds is relative to the ground. The eagle emits a shrill cry whose frequency is 3240 Hz. The speed of sound is 330 m/s. What frequency do you hear?

Explanation / Answer

1) y(x,t)=0.17sin(2.7*pi*t+0.55*pi*x)

here v=w/k=lambda*frequency

We know the general expression for a traveling wave is y(x,t)=Amplitude*sin(kx-wt)=Amplitude*sin(2*pi*x/lambda-2*pi*frequency*t); therefore

2*pi/lambda=0.55*pi ---> lambda=2/0.55=3.64 m

2*pi*frequency=2.7*pi ---> frequency=2.7/2=1.35 Hz

v=3.64m*1.35 Hz=4.91m/s

2)

The time difference between first and second echo is

t1 = 2(x2)/v - 2(x1)/v =1.88 s

third echo travels an additional distance of 2x

after two echos has occured

t2 = 2(x1)/v = 1.69

x1 = vt2/2

from the first equation

x2 - x1 = vt1/2

x2 = v(t1)/2 + x1 = v(t1)/2 + v(t2)/2 = v/2( t1+ t2)

distance between cliffs d = x1 + x2

d = v(t2)/2 + v/2 ( t1 + t2) = v/2 ( t1 + 2(t2))

= 343/2 ( 1.88 + 2( 1.69) = 902.09 m

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