Topic 3.4: Standing Waves and Resonance Questions

State what is meant by **resonance**, and state which harmonic is called the **fundamental**.

Describe two ways in which a standing wave differs from a travelling wave.

State what is meant by a node on a standing wave.

On a vibrating guitar string a standing wave is photographed.

The distance from one node to the next node along the string is measured as 0.18 m.

Determine the wavelength of the wave on the string.

A standing wave is set up on a stretched wire.

The distance between two neighbouring **nodes** is measured as 12 cm.

The speed of the wave on the wire is 480 m s⁻¹.

Determine the wavelength of the wave and its frequency.

A standing wave is set up on a stretched string fixed at both ends.

Outline how this standing wave is formed.

An organ pipe of length 0.250 m is closed at one end and open at the other.

The speed of sound inside the pipe is 340 m s⁻¹.

What are the frequencies of the first two harmonics (the two lowest resonant frequencies) of the pipe?

Two points P and Q lie on a string carrying a standing wave.

They are in the same loop, between the same two nodes, and neither point is at a node.

Identify the phase difference between P and Q, and compare this with two points the same distance apart on a travelling wave.

A standing wave on a stretched string has neighbouring nodes 0.12 m apart, and the string vibrates with a frequency of 40 Hz.

Calculate the speed of the wave on the string.

A wire of length 0.35 m is fixed at both ends and a wave travels along it at 140 m s⁻¹.

Show that its fundamental frequency is 200 Hz, and hence determine the frequency of its **third** harmonic.

A guitarist tunes a string so its fundamental is 256 Hz.

The wave speed on the string is 440 m s⁻¹.

Estimate the vibrating length of the string, and identify how many nodes (including the two ends) appear when it vibrates in its **third** harmonic.

A microwave source faces a metal reflector, setting up a standing wave between them.

A small detector finds that neighbouring points of **minimum** signal are 5.0 cm apart.

Microwaves travel at c = 3.0 × 10⁸ m s⁻¹.

Determine the frequency of the microwaves.

A string is fixed at both ends and made to vibrate so that a standing wave forms along it.

Outline how the standing wave is produced, and explain why some points on the string (the nodes) never move while others (the antinodes) move the most.

A standing wave is formed by two identical waves travelling in opposite directions, each of amplitude 1.5 cm.

Show that the amplitude of the string at an antinode is 3.0 cm.

A pipe **open at both ends** sounds its fundamental at 170 Hz.

The speed of sound in the air is 340 m s⁻¹.

Deduce the length of the pipe.

An organ pipe of length 0.50 m is open at both ends.

The speed of sound in the air column is 340 m s⁻¹.

Calculate the frequency of the fundamental and the frequency of the second harmonic, and state the wavelength condition you used.

A pipe of length 0.17 m is **closed at one end** and open at the other.

The speed of sound in the air column is 340 m s⁻¹.

Calculate the frequency of the fundamental and the frequency of the next harmonic above it.

In a microwave oven with the turntable removed, a flat tray of melted-cheese spots forms an evenly spaced row.

Neighbouring melted spots are measured to be 5.9 cm apart.

The speed of microwaves is 3.0 × 10⁸ m s⁻¹.

Deduce the frequency of the microwaves.

Explain why a pipe closed at one end can resonate at 1500 Hz and at 2500 Hz but **not** at 2000 Hz, given that its fundamental is at 500 Hz.

A student sets up a standing wave on a 1.2 m wire fixed at both ends and observes a pattern with exactly three loops (the third harmonic).

Determine the wavelength of the wave on the wire, the distance between two neighbouring nodes, and — given the wave speed on the wire is 90 m s⁻¹ — the frequency of this harmonic.