Question

Directions: Please answer the following question in complete sentences. Be sure to label all geometric objects in any illustrations (if any). I will accept an answer in a scanned image format, as a pdf, or as a picture taken and sent from your awesome smart phone. Question: Sound is a result of waves applying pressure to a outward in a spherical shape, the trigonometric function person's eardrum. For a pure sound wave radiating 2r P -ct COS can be used to model the sound pressure at a radius of r feet from the source, wheret is time in seconds, A is length of the sound wave in feet, c is speed of sound in feet per second, and a is the maximum sound pressure at the source measured in pounds per square foot. Let A4.9 ft and c 1026 ft per sec. ( (a) Let a 0.4 lb per ft2. Graph the sound pressure at distance r 10 ft from its source for 0 sts0.05. Describe P at this distance. (b) Now let a 3 and t 10. Graph the sound pressure for 0s r20. Describe the how the pressure P changes as the radius goes fromr-0 to r20. nA, where n is a positive integer. Use the difference identity (c) Suppose a person stands at a radius r so for cosine to simplify P in this situation. that r

Answer 1

There is a slight error in the formula for pressure. It is dimensionally inconsistent. The argument of cosine can't have any dimension. Moreover, two different dimensioned quantities are being added. The error is probably in placing the brackets.

2T -(r ct) P =cos COs

(a)

0.04 0.03 0.02 F 0.01 0 -0.01 -0.02 -0.03 -0.04 0.02 0,025 0.04 0 0.005 0.01 0.015 0.03 0.035 0.045 0.05

Sinusoidal variation. As expected, the average (over time) sound level is constant at a fixed distance from the source.

(b)

Because the magnitude of pressure near the source is infinitely large, I have plotted the pressure for

T E0.5, 20

0.5 0 -0.5 1 -1.5 4 10 12 16 2 6 8 14 18 20 co

As expected, the intensity of sound level decreases from the distance at a fixed time.

(c)

27T (nA-ct) P(r n, t) COS nA

$P(n\lambda,t)=\frac{a}{n\lambda}\cos\left (2\pi n-\omega t\right )$

where

$\omega=\frac{2\pi c}{\lambda}$

so that

$P(n\lambda,t)=\frac{a}{n\lambda}\cos\left (\omega t\right )$

There no expansion needed actually. Just the fact that

$\cos\left (2\pi-\theta\right )=\cos\theta$