Standing Waves and Resonance: notes and practice questions

A flute is modelled as a pipe of length $L$ that is open at both ends.

What is the wavelength of the fourth-harmonic standing wave in the flute?

A. $2L$

B. $L$

C. $\frac{L}{2}$

D. $\frac{L}{4}$

A third-harmonic standing wave is formed on a string of length $60\text{ cm}$ that is fixed at both ends.

What two points along the string have a phase difference of $\pi$?

A. $10\text{ cm}$ and $50\text{ cm}$

B. $5\text{ cm}$ and $15\text{ cm}$

C. $15\text{ cm}$ and $25\text{ cm}$

D. $25\text{ cm}$ and $35\text{ cm}$

A standing wave is formed on a string of length $L$ that is fixed at both ends. The standing wave has three antinodes. What is the wavelength of the standing wave?

A. $\frac{L}{3}$

B. $\frac{2L}{3}$

C. $L$

D. $\frac{3L}{2}$

A student performs an experiment with a resonating air column, using a tube closed at one end. The effective length of the air column is $0.60\text{ m}$. The speed of sound in the air inside the tube is measured to be $340\text{ m s}^{-1}$.

What are the frequencies of the first two harmonics that can be produced in this tube?

A. $142\text{ Hz}$ and $283\text{ Hz}$

B. $142\text{ Hz}$ and $425\text{ Hz}$

C. $283\text{ Hz}$ and $567\text{ Hz}$

D. $283\text{ Hz}$ and $850\text{ Hz}$

A musician designs two wind instruments. The first instrument is a flute, effectively a pipe open at both ends, with a length of $L_F$. Its fundamental (first harmonic) frequency is $f_0$. The second instrument is a pan flute pipe, effectively open at one end and closed at the other, designed to produce the same fundamental frequency $f_0$.

If the length of the flute (open at both ends) is $0.72 \text{ m}$, what is the length of the pan flute pipe (open at one end and closed at the other)?

A. $0.18 \text{ m}$

B. $0.36 \text{ m}$

C. $0.72 \text{ m}$

D. $1.44 \text{ m}$

A car's suspension system is designed to be heavily damped. The car travels over a series of bumps on a road. The frequency of the bumps, $f$, depends on the speed of the car. The natural frequency of the car's vertical oscillation is $f_0$. Which graph best shows the variation of the amplitude of the car's vertical oscillations with the frequency $f$?

A standing sound wave is formed in a pipe of length $L$ that is closed at one end. The standing wave has two nodes. What is the wavelength of the standing wave?

A. $\frac{3L}{4}$

B. $\frac{2L}{3}$

C. $L$

D. $\frac{4L}{3}$

The absorption of infrared radiation by greenhouse gases is a key part of the greenhouse effect. This absorption is particularly effective because

A. the greenhouse gases are at a much higher temperature than the Earth's surface.

B. the infrared radiation causes the gas molecules to undergo nuclear fission.

C. the frequency of the infrared radiation matches the natural vibrational frequency of the greenhouse gas molecules.

D. the greenhouse gas molecules are much larger than the other molecules in the atmosphere.

An organ pipe that is closed at one end resonates in its first overtone with a frequency of $450 \text{ Hz}$.

What are two other resonant frequencies, in $\text{Hz}$, for this pipe?

A. $150$ and $750$

B. $225$ and $675$

C. $300$ and $600$

D. $150$ and $300$

An organ pipe is closed at one end. It resonates in the fifth harmonic with a frequency of $450 \text{ Hz}$.

What are two other harmonic frequencies, in Hz, at which this pipe can resonate?

A. $180$ and $270$

B. $90$ and $270$

C. $150$ and $300$

D. $90$ and $180$

A string of length $L$ is fixed at both ends and vibrates in its second harmonic. What is the phase difference between a point on the string at a distance of $\frac{L}{8}$ from one end and a point at a distance of $\frac{3L}{8}$ from the same end?

A. $0$

B. $\frac{\pi}{4}$

C. $\frac{\pi}{2}$

D. $\pi$