Physics: Standing Waves and Resonance Worksheet

Question 1

A string is fixed at both ends and vibrates in its fundamental mode. The length of the string is 1.20 m. What is the wavelength of the standing wave?

Accepted Answer

The wavelength is 2.40 m. For the fundamental mode on a string fixed at both ends, the string length is half a wavelength, so wavelength = 2L = 2(1.20 m) = 2.40 m.

Question 2

A wave travels on a string at 96 m/s. The string is 0.80 m long and fixed at both ends. What is the fundamental frequency?

Accepted Answer

The fundamental frequency is 60 Hz. For a string fixed at both ends, f1 = v/(2L) = 96 m/s divided by 1.60 m = 60 Hz.

Question 3

A 2.0 m string fixed at both ends has a wave speed of 120 m/s. Find the frequency of the third harmonic.

Accepted Answer

The third harmonic frequency is 90 Hz. The fundamental frequency is f1 = v/(2L) = 120 m/s divided by 4.0 m = 30 Hz, so f3 = 3f1 = 90 Hz.

Question 4

A standing wave on a string fixed at both ends has 4 loops. What harmonic number is this, and how many nodes are present including the endpoints?

Accepted Answer

This is the fourth harmonic, and it has 5 nodes including the endpoints. On a string fixed at both ends, the number of loops equals the harmonic number, and the number of nodes is one more than the number of loops.

Question 5

A student observes a standing wave with nodes spaced 0.25 m apart along a rope. What is the wavelength of the wave?

Accepted Answer

The wavelength is 0.50 m. Adjacent nodes are separated by half a wavelength, so wavelength = 2(0.25 m) = 0.50 m.

Question 6

A string produces a fundamental frequency of 110 Hz. What are the frequencies of the second, third, and fourth harmonics?

Accepted Answer

The second harmonic is 220 Hz, the third harmonic is 330 Hz, and the fourth harmonic is 440 Hz. Harmonics on a string fixed at both ends are whole-number multiples of the fundamental frequency.

Question 7

An open-open pipe is 0.50 m long. The speed of sound in air is 343 m/s. What is the fundamental frequency of the pipe?

Accepted Answer

The fundamental frequency is 343 Hz. For an open-open pipe, f1 = v/(2L) = 343 m/s divided by 1.00 m = 343 Hz.

Question 8

A closed-open pipe has a length of 0.85 m. The speed of sound is 340 m/s. What is its fundamental frequency?

Accepted Answer

The fundamental frequency is 100 Hz. For a closed-open pipe, f1 = v/(4L) = 340 m/s divided by 3.40 m = 100 Hz.

Question 9

A closed-open pipe has a fundamental frequency of 85 Hz. Which of these frequencies are resonant frequencies for the pipe: 170 Hz, 255 Hz, 340 Hz, and 425 Hz?

Accepted Answer

The resonant frequencies are 255 Hz and 425 Hz. A closed-open pipe supports only odd harmonics, so the allowed frequencies are 1f1, 3f1, 5f1, and so on.

Question 10

A tuning fork vibrates at 256 Hz and causes a nearby air column to vibrate strongly. What is this strong response called, and why does it happen?

Accepted Answer

The strong response is called resonance. It happens because the tuning fork frequency matches or is very close to one of the natural frequencies of the air column, causing energy transfer to build a large vibration.

Question 11

A standing wave is formed by two identical waves traveling in opposite directions. Each wave has a frequency of 20 Hz and a wavelength of 0.60 m. What is the wave speed?

Accepted Answer

The wave speed is 12 m/s. The speed is v = f times wavelength = 20 Hz times 0.60 m = 12 m/s.

Question 12

On a standing wave diagram, a point stays at rest while nearby parts of the medium move up and down. What is this point called? Explain what causes it.

Accepted Answer

The point is called a node. It occurs where the two interfering waves always cancel each other, producing destructive interference at that location.

Question 13

A 1.5 m string fixed at both ends is vibrating in the second harmonic. What is the wavelength of this standing wave?

Accepted Answer

The wavelength is 1.5 m. For a string fixed at both ends, L = n(wavelength)/2. With n = 2, L = wavelength, so the wavelength equals 1.5 m.

Question 14

A guitar string has a fundamental frequency of 196 Hz. If a guitarist lightly touches the midpoint of the string and plucks it, the string vibrates mainly in the second harmonic. What frequency is produced?

Accepted Answer

The frequency produced is 392 Hz. The second harmonic is twice the fundamental frequency, so f2 = 2(196 Hz) = 392 Hz.

Question 15

Two students are testing a rope. Student A shakes the rope at a random frequency and sees only small motion. Student B shakes the rope at a frequency that produces a clear standing wave with large antinodes. Which student is driving the rope closer to resonance, and how do you know?

Accepted Answer

Student B is driving the rope closer to resonance. A clear standing wave with large antinodes shows that the driving frequency is close to one of the rope's natural frequencies, so energy builds up in the pattern.

Physics: Standing Waves and Resonance

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