Animated Solution for Physics - Waves: Two cars are approaching each other at an equal speed of $7.2\text{ km/h}$. When they see each other, both blow horns having frequency of $676\text{ Hz}$. The beat frequency heard by each driver will be ......... $\text{Hz}$. [Velocity of sound in air is $340\text{ m/s}$.]

Imagine you are sitting in Car A, driving down a straight road at a speed of $7.2\text{ km/h}$. Coming directly towards you in the opposite lane is Car B, also traveling at $7.2\text{ km/h}$. Suddenly, both of you press your horns simultaneously. The horns are identical, each emitting a sound wave with a frequency of $676\text{ Hz}$.

Before we dive into the physics of sound, let's make sure our units are consistent. The speed of sound is given in meters per second ($340\text{ m/s}$), so we must convert the speed of the cars from $\text{km/h}$ to $\text{m/s}$.

Now, both cars are moving at a neat $2\text{ m/s}$.

The Doppler Effect

Because both the source of the sound (the other car) and the observer (you) are in motion relative to the medium (air), we must use the general Doppler effect formula:

Here, $v$ is the speed of sound, $v_o$ is the speed of the observer, and $v_s$ is the speed of the source.

Sign Convention is crucial here. Since you (the observer) are moving towards the source, you encounter more wave crests per second, which increases the apparent frequency. Therefore, we use a plus sign in the numerator. Simultaneously, the source is moving towards you, compressing the wavefronts and further increasing the frequency. Therefore, we use a minus sign in the denominator.

Let's substitute our known values into the Doppler equation to find the frequency of Car B's horn as heard by you in Car A:

Look closely at the numbers! The examiners have been kind to us. Notice that $676$ is exactly twice $338$. This makes our calculation incredibly elegant:

So, the horn from the approaching car sounds higher pitched to you, at $684\text{ Hz}$.

But the question doesn't just ask for the apparent frequency; it asks for the beat frequency.

While you are hearing the other car's horn at $684\text{ Hz}$, you are also hearing your own car's horn. Since you are stationary relative to your own horn, there is no Doppler shift for it. You hear it at its true frequency of $676\text{ Hz}$.

When two sound waves of slightly different frequencies reach your ear simultaneously, they interfere with each other, creating a pulsating sound known as beats. The beat frequency is simply the absolute difference between the two frequencies:

Let's find that difference:

Each driver will hear a distinct throbbing sound pulsating $8$ times every second. The physics of relative motion and wave interference beautifully combine to give us our final answer of $8\text{ Hz}$.