Animated Solution for Physics - Waves: Two sources of sound $S_1$ and $S_2$ produce sound waves of same frequency $660\text{ Hz}$. A listener is moving from source $S_1$ towards $S_2$ with a constant speed $u\text{ m/s}$ and he hears $10\text{ beats/s}$. The velocity of sound is $330\text{ m/s}$. Then, $u$ equal to

Walking Between Speakers

Imagine you are standing in an open field exactly midway between two massive concert speakers, $S_1$ and $S_2$. Both speakers are emitting a pure, continuous tone at exactly $660\text{ Hz}$. If you stand perfectly still, you hear a single, steady pitch. But the moment you start walking towards one of the speakers, the physics of the situation changes dramatically.

In this problem, you are moving away from $S_1$ and towards $S_2$ with a constant speed $u$. Because of your motion, the sound waves from the two speakers interact with your ears differently. This relative motion gives rise to two of the most fascinating phenomena in wave mechanics: the Doppler Effect and Acoustic Beats.

The Doppler Effect dictates that the frequency of a wave changes if there is relative motion between the source and the observer. Think of it like riding a boat on a lake. If you drive the boat into the oncoming waves, you hit them more frequently. If you drive away from the waves, they take longer to catch up to you, so you hit them less frequently.

As you walk away from speaker $S_1$, the sound waves have to "chase" you. Consequently, the frequency you perceive is lower than the actual frequency emitted. The formula for an observer moving away from a stationary source is:

Conversely, as you walk towards speaker $S_2$, you are rushing headlong into its sound waves. You encounter the wave crests more frequently, meaning the perceived frequency is higher. The formula for an observer moving towards a stationary source is:

Here, $v$ is the speed of sound in air ($330\text{ m/s}$), and $u$ is your walking speed.

Because you are moving, your left ear (facing $S_1$) and your right ear (facing $S_2$) are effectively receiving two slightly different frequencies, $f_{S_1}^{\prime }$ and $f_{S_2}^{\prime }$. When two sound waves of slightly different frequencies interfere, they periodically align (constructive interference) and misalign (destructive interference).

This creates a pulsing sound—a "wah-wah-wah" effect—known as beats. The number of pulses you hear per second is called the beat frequency ($f_b$), which is simply the absolute difference between the two perceived frequencies:

Now, let's substitute our Doppler expressions into the beat frequency equation to see the underlying mathematical structure:

We can factor out the common term $\frac{f}{v}$:

Notice the beautiful algebraic cancellation that happens inside the brackets. The $v$ terms annihilate each other ($v-v=0$), while the $u$ terms reinforce each other ($u-(-u)=2u$). This leaves us with a highly elegant, simplified expression for the beat frequency:

We are given all the necessary parameters to solve for your walking speed $u$. We know the beat frequency $f_b=10\text{ Hz}$, the source frequency $f=660\text{ Hz}$, and the speed of sound $v=330\text{ m/s}$. Let's plug these numbers into our master equation:

The fraction $\frac{660}{330}$ simplifies perfectly to $2$.

Dividing both sides by $4$, we find:

So, you are walking at a brisk pace of $2.5\text{ m/s}$ to hear exactly 10 beats per second. The interplay of relative motion and wave interference gives us a precise tool to measure velocity!