Animated Solution for Physics - Waves: A string 2.0 m long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is

Standing waves on a string are one of the most beautiful and visual phenomena in physics. When a string is fixed at both ends and plucked or driven by a vibrator, waves travel back and forth, reflecting off the boundaries. At certain specific frequencies, these traveling waves interfere perfectly to create a stationary pattern known as a standing wave. Let's dive into the mechanics of this problem to uncover the wave speed and the fundamental frequency.

Imagine a string stretched between two rigid supports, exactly $2.0\text{ m}$ apart. We are told that a vibrator is driving this string at a frequency of $240\text{ Hz}$, causing it to vibrate in its third harmonic mode.

What does the third harmonic look like? Because the string is fixed at both ends, those ends must be nodes—points of zero displacement. In the third harmonic, the string forms exactly three distinct loops (or antinodes) between the supports. This means there are two additional nodes spaced evenly along the string.

To connect the physical dimensions of the string to the frequency of its vibration, we use the master equation for the frequency of the $n^{\text{th}}$ harmonic of a string fixed at both ends:

Here, $f_n$ is the frequency of the $n^{\text{th}}$ harmonic, $n$ is the harmonic number (which corresponds to the number of loops), $v$ is the speed of the wave traveling along the string, and $L$ is the total length of the string.

We have all the pieces of the puzzle. We know it's the third harmonic, so $n=3$. The frequency $f_3$ is $240\text{ Hz}$, and the length $L$ is $2.0\text{ m}$. Let's substitute these values into our equation:

Simplifying the denominator, we get:

Now, we isolate the wave speed $v$:

Dividing $240$ by $3$ gives $80$, and multiplying that by $4$ yields a wave speed of:

But we aren't done yet! The question also asks for the fundamental frequency, $f_1$. The fundamental frequency is the lowest possible frequency at which the string can form a standing wave (a single loop, $n=1$).

The beautiful property of harmonics is that they are integer multiples of the fundamental frequency:

Therefore, to find the fundamental frequency, we simply divide the third harmonic frequency by $3$:

We have successfully found both values: the wave speed is $320\text{ m/s}$ and the fundamental frequency is $80\text{ Hz}$. This perfectly matches option (b).