physics questions?

Question

1.) What is the wavelength of a 440 Hz sound wave in the air?

2.) While riding in your boat, you notice one wave pass by every two seconds. If the actual frequency of the waves is 1 Hz, with what speed are you moving through the water? The speed of the waves is 5 m/s. Are you moving towards the waves or away?

Answer 1

1. Divide the speed of sound in air (given by various references as 1070 fps, 1100 fps, etc.) by 440.

1070/440 = ~2.432 feet

2. First, the easy part. You are moving away from the waves. If you were sitting still, one wave would catch you every second. But since you're "running away", each wave takes two seconds to reach you.

Now, the harder part. Let's call F the real wave frequency (1 Hz), and Fa the apparent wave frequency (0.5 Hz). Vw is the velocity of the wave, and Vb is the velocity of the boat. Do you see that if Vb = 0, then Fa = F? Good. Now consider that if Vb = Vw, then Fa = 0. You would just exactly keep pace with the waves, and they'd never catch you. So,
Fa = F(Vw - Vb)/Vw
Since Fa = 0.5,
0.5 = 1(5 - Vb)/5
2.5 = 5 - Vb
2.5 + Vb = 5
Vb = 2.5
So the boat is moving 2.5 m/s away from the waves.