Stationary Waves

The Behaviour of Waves

Define a stationary wave.

A wave in which oscillations occur in fixed positions, with nodes and antinodes that remain in the same locations.

What is another name for a stationary wave?

A standing wave.

How is a stationary wave formed?

By the superposition of two coherent waves of the same frequency travelling in opposite directions.

What conditions are needed to form a stationary wave?

Two waves with the same frequency, similar amplitudes, travelling in opposite directions and having a constant phase relationship.

Define coherent waves.

Waves that have the same frequency and a constant phase relationship.

Why must the waves forming a stationary wave be coherent?

So that their interference pattern remains stable and the nodes and antinodes stay in fixed positions.

What is a progressive wave?

A wave that transfers energy from one place to another.

What is the main difference between a stationary wave and a progressive wave?

A progressive wave transfers energy, whereas a stationary wave has no net transfer of energy along the wave.

Why is a stationary wave called stationary?

Its overall wave pattern does not travel along the medium; the nodes and antinodes remain in fixed positions.

How can a stationary wave be produced on a string?

A travelling wave reflects from a fixed end and superposes with the incident wave travelling in the opposite direction.

Define a node.

A point on a stationary wave where the amplitude of oscillation is always zero.

Define an antinode.

A point on a stationary wave where the amplitude of oscillation is maximum.

What is the amplitude at a node?

Zero.

Where is the amplitude greatest in a stationary wave?

At an antinode.

How does amplitude vary between a node and an antinode?

It increases from zero at the node to a maximum at the antinode.

What is the distance between two adjacent nodes?

λ/2.

What is the distance between two adjacent antinodes?

λ/2.

What is the distance between a node and the nearest antinode?

λ/4.

What is the phase relationship between points between the same two adjacent nodes?

They oscillate in phase.

What is the phase difference between particles in adjacent loops of a stationary wave?

180° or π radians.

What happens at the fixed ends of a vibrating string?

The fixed ends must be nodes.

What determines which stationary wave patterns can form on a string fixed at both ends?

Only wavelengths that allow nodes at both fixed ends can form.

Define the fundamental mode.

The lowest-frequency stationary wave pattern that can form on a system.

What is another name for the fundamental mode?

The first harmonic.

How many antinodes are present in the fundamental mode of a string fixed at both ends?

One.

How many nodes are present in the fundamental mode of a string fixed at both ends, including the ends?

Two.

What is the relationship between string length and wavelength for the fundamental mode?

L = λ/2.

What is the wavelength of the fundamental mode for a string of length L?

λ = 2L.

What is the first overtone?

The second harmonic.

How many half-wavelengths fit into the string in the first overtone?

Two.

What is the wavelength of the first overtone?

λ = L.

What is the frequency of the first overtone compared with the fundamental frequency?

2f₀.

What is the second overtone?

The third harmonic.

What is the wavelength of the second overtone?

λ = 2L/3.

What is the frequency of the second overtone compared with the fundamental frequency?

3f₀.

What is the third overtone?

The fourth harmonic.

What is the wavelength of the third overtone?

λ = L/2.

What is the frequency of the third overtone compared with the fundamental frequency?

4f₀.

What is the general wavelength equation for the nth harmonic on a string fixed at both ends?

λₙ = 2L/n.

What is the general frequency equation for the nth harmonic?

fₙ = nf₀.

What is the relationship between harmonic number and number of antinodes?

The nth harmonic has n antinodes.

What is the relationship between harmonic number and number of nodes on a string fixed at both ends?

The nth harmonic has n + 1 nodes, including the two fixed ends.

What equation relates wave speed, frequency and wavelength?

v = fλ.

How can the fundamental frequency be calculated from wave speed and string length?

f₀ = v/(2L).

What factors affect the speed of a transverse wave on a string?

The tension in the string and its mass per unit length.

Define mass per unit length.

The mass of a string divided by its length.

What is the symbol for mass per unit length?

μ.

What are the units of mass per unit length?

kg m⁻¹.

What equation gives the speed of a transverse wave on a string?

v = √(T/μ).

What does T represent in the string wave-speed equation?

The tension in the string, measured in newtons (N).

What does μ represent in the string wave-speed equation?

The mass per unit length of the string, measured in kg m⁻¹.

How does increasing tension affect wave speed on a string?

Wave speed increases because v ∝ √T.

How does increasing mass per unit length affect wave speed?

Wave speed decreases because v ∝ 1/√μ.

What equation gives the fundamental frequency of a stretched string?

f₀ = (1/2L)√(T/μ).

How does fundamental frequency depend on string length?

f₀ ∝ 1/L.

What happens to fundamental frequency if the string length is doubled while other factors stay constant?

It halves.

How does fundamental frequency depend on tension?

f₀ ∝ √T.

What happens to fundamental frequency if the tension is quadrupled?

It doubles.

How does fundamental frequency depend on mass per unit length?

f₀ ∝ 1/√μ.

What happens to fundamental frequency if mass per unit length is quadrupled?

It halves.

How can the relationship f₀ ∝ 1/L be investigated experimentally?

Vary the vibrating length of the same string while keeping tension and mass per unit length constant, then measure the fundamental frequency.

What graph verifies that f₀ ∝ 1/L?

A graph of f₀ against 1/L should be a straight line through the origin.

How can the relationship f₀ ∝ √T be investigated experimentally?

Keep string length and mass per unit length constant, vary the tension using hanging masses, and measure the fundamental frequency.

What graph verifies that f₀ ∝ √T?

A graph of f₀ against √T should be a straight line through the origin.

How can the relationship f₀ ∝ 1/√μ be investigated experimentally?

Use strings with different mass per unit lengths while keeping length and tension constant, then measure the fundamental frequency.

What graph verifies that f₀ ∝ 1/√μ?

A graph of f₀ against 1/√μ should be a straight line through the origin.

How can mass per unit length be measured experimentally?

Measure the mass and total length of the string and calculate μ = mass/length.

What is a sonometer?

An apparatus used to investigate the frequencies of vibrations of a stretched string under tension.

How can the frequency of a vibrating sonometer string be measured?

Using a microphone connected to an oscilloscope or data logger.

How is tension commonly varied in a sonometer experiment?

By changing the hanging mass attached to the string.

How is the tension produced by a hanging mass calculated?

T = mg.

Why should only one variable be changed at a time in a sonometer experiment?

To determine the effect of that variable while keeping the others controlled.

What safety precaution should be used when experimenting with stretched wires?

Wear eye protection in case the wire snaps.

Why should a drop box be placed below large hanging masses?

To reduce the risk of injury or damage if the masses fall.

A string of length 0.65 m vibrates in its fundamental mode. What is its wavelength?

λ = 2L = 2 × 0.65 = 1.30 m.

A string has a fundamental wavelength of 1.30 m and wave speed of 362 m s⁻¹. What is its frequency?

f = v/λ = 362/1.30 = 278 Hz.

If the wavelength of a stationary wave decreases while wave speed stays constant, what happens to its frequency?

The frequency increases because f = v/λ.

If two complete wavelengths fit into a string of length L, what is the wavelength?

λ = L/2.

If two complete wavelengths fit into the same string instead of half a wavelength, how does the frequency compare?

It is four times the fundamental frequency.

Why are only certain frequencies possible on a string fixed at both ends?

The string must have nodes at both ends, so only wavelengths that fit an exact number of half-wavelengths into the string are allowed.