1. A sound is made by a speaker, which then radiates in all directions. The sound wave

is described by
A(r) sin(2r - 600t)
where the distances are in meters, and the time in seconds. The A(r) just indicates that
the amplitude has some distance dependence--it decreases smoothly as the wave gets
farther from the source

a) What is the velocity of this wave?

b) How long does it take for the sound to travel from the speaker to a listener 100 m
away (the length of a football field)?

velocity?

freq*lambda=velocity

freq= 600/2PI

lambda= 2PI/2=PI

velocity= 600/2PI * PI= 300m/s

To find the velocity of the wave, we need to determine the relationship between distance and time in the given wave equation. In this case, the wave equation is given as A(r) sin(2r - 600t), where r represents distance in meters and t represents time in seconds.

a) Velocity of the wave:
The general equation for a wave is given by v = λ*f, where v is the velocity, λ is the wavelength, and f is the frequency.

In our wave equation A(r) sin(2r - 600t), we can see that the distance (r) is changing with time (t), which means the wave is not at a constant wavelength. Therefore, we cannot directly find the velocity using the standard wave equation.

To find the velocity, we need to calculate the derivative of the distance function with respect to time. In this case, we take the derivative of (2r - 600t) with respect to t.

d(2r - 600t)/dt = -600

The negative sign indicates that the wave is traveling in the opposite direction to the negative direction of the time axis. So, the velocity of the wave is 600 m/s in the opposite direction of decreasing time.

b) To calculate the time it takes for sound to travel from the speaker to a listener 100 m away, we need to find the time at which the distance (r) is equal to 100 m.

Given the equation A(r) sin(2r - 600t), we need to set r to 100 and solve for t.

100 = 2r - 600t

Rearranging the equation, we have:

600t = 2r - 100

600t = 2(100) - 100
600t = 200 - 100
600t = 100
t = 100/600
t = 1/6

Therefore, it takes 1/6 seconds or approximately 0.1667 seconds for the sound to travel from the speaker to the listener 100 m away.