Looking at longitudinal waves
Section: Waves | Syllabus: Cambridge AS Level Physics (9702)
Longitudinal Waves
Sound waves travel differently from transverse waves. In a longitudinal wave, the particles of the medium oscillate in a direction parallel to the direction of energy transfer.
The terms amplitude, speed, wavelength, frequency, phase and intensity that apply to transverse waves can all be used to describe longitudinal waves. The wave equation v = fλ also applies.
Longitudinal Wave A progressive wave in which the particles of the medium oscillate in a direction parallel to the direction of energy transfer.The Slinky Spring Demonstration
You can make a longitudinal wave using a "slinky" spring:
- Place the spring straight on the ground
- Fix one end of the spring (or have someone hold it)
- Push and pull the other end rapidly
You will see waves of compression (where the turns of the coil are pushed close together) moving along the spring, but there is no motion from side to side.
Compression Regions where particles are closer together than normal, modelling regions of high pressure in a sound wave. Rarefaction Regions between compressions where particles are further apart than normal, modelling regions of low pressure. FIG 7.3: Longitudinal Wave StructureShow a longitudinal wave (such as a sound wave or slinky spring) with particles represented as dots. Clearly label: compressions (C) where particles are close together, rarefactions (R) where particles are spread apart, wavelength (λ) measured from compression to compression, and direction of wave propagation. Show the direction of particle oscillation as parallel to wave direction.
Wavelength MeasurementThe wavelength of a longitudinal wave is measured between the centre of one compression and the centre of the next compression (or between two successive rarefactions).
Sound Waves as Longitudinal Pressure Waves
Sound waves travel as longitudinal pressure waves through gases, liquids and solids. The denser the medium, the faster the sound wave travels.
For a sound wave moving through air, the atmospheric pressure varies along the wave:
- Compression (C): Higher than atmospheric pressure
- Rarefaction (R): Lower than atmospheric pressure
- The variation in atmospheric pressure of many sound waves is only about 0.03%
A displacement-time graph for an oscillating particle in a longitudinal wave will be a sine curve, similar to a transverse wave. The forward-and-back motion is represented by up and down on the graph.
Amplitude in Longitudinal WavesA longitudinal wave with greater amplitude will have higher magnitude compressions (particles pushed closer together, higher pressure) and greater magnitude rarefactions (particles further apart, lower pressure). Higher amplitude means higher intensity and louder sound.
Obtaining a Graphical Representation of Sound Waves
The longitudinal nature of sound waves makes them difficult to draw or measure directly. However, we can use a microphone or signal generator, together with a cathode-ray oscilloscope (CRO), to display sound waves in a transverse form.
The Microphone
A microphone converts sound waves into varying electrical signals. The pressure wave is converted into a varying voltage.
The Signal Generator
To produce sound waves containing only one frequency (a pure note), we use a signal generator and a loudspeaker. A signal generator has an adjustable frequency output and amplitude control. The sine wave setting produces pure tones.
The Loudspeaker
A loudspeaker does the opposite of a microphone-it converts electrical signals into sound waves. A cone vibrates with the same frequency as the input electrical wave, producing sound waves in air.
The Cathode-Ray Oscilloscope (CRO)
A cathode-ray oscilloscope is a device for displaying electrical signals on a screen. The two most important controls are:
Time-base Controls how quickly the trace moves across the screen. Setting is stated in milliseconds per division or microseconds per division (ms div⁻¹ or μs div⁻¹). Y-gain (or Gain) Controls the height of the displayed wave. Setting is stated in volts per division or millivolts per division (V div⁻¹ or mV div⁻¹).With no input and time-base off, a single spot is seen. When the time-base is on, the spot moves across the screen from left to right, appearing as a straight line or 'trace'.
When an electrical input is connected (from a microphone or signal generator), the input voltage causes the spot to be displaced vertically. The display now resembles a graph of voltage against time.
FIG 7.4: Cathode-Ray Oscilloscope DisplayShow an oscilloscope screen displaying a sinusoidal wave trace. Label: the grid divisions, time-base setting (e.g., 2 ms/div on horizontal axis), Y-gain/voltage setting (e.g., 0.5 V/div on vertical axis), period (T) measured horizontally between two peaks, and amplitude measured vertically from centre line to peak. Include a representation of the oscilloscope controls.
ImportantThe distance between two peaks on the oscilloscope display gives the period of the wave, not the wavelength. The amplitude shown is the amplitude of the input electrical signal in volts, not a direct measurement of the sound wave amplitude.
Using an Oscilloscope to Find Frequency
Connect a microphone to an oscilloscope to display the characteristics of a sound wave. The screen displays a voltage-time graph representing the sound wave.
To find frequency:
- Adjust the time-base so only 2-3 peaks are displayed
- Measure the period T using the time-base setting
- Calculate frequency: f = 1/T
Question: An oscilloscope display shows a sine wave. The time-base setting is 0.2 ms div⁻¹. Two complete waves span exactly 10 horizontal scale divisions. The speed of sound in air is 340 m s⁻¹. (a) Determine the frequency of the sound. (b) Calculate the wavelength.
Solution(a) Period of one wave = ½ × 0.2 ms × 10 = 1.0 ms = 1.0 × 10⁻³ s
f = 1/T = 1/(1.0 × 10⁻³) = 1000 Hz
(b) λ = v/f = 340/1000 = 0.34 m
Worked Example: Reading CRO SettingsQuestion: An oscilloscope has time-base set to 2 ms div⁻¹ and y-gain set to 0.5 V div⁻¹. A wave displayed spans 4 divisions for one complete cycle and has a peak height of 3 divisions from the centre line. Calculate: (a) the amplitude, (b) the frequency.
Solution(a) Amplitude = 3 × 0.5 = 1.5 V
(b) Period T = 4 × 2 ms = 8 ms = 8 × 10⁻³ s
f = 1/T = 1/(8 × 10⁻³) = 125 Hz
Comparing Transverse and Longitudinal Waves
| Feature | Transverse Waves | Longitudinal Waves |
|---|---|---|
| Direction of oscillation | Perpendicular to wave travel | Parallel to wave travel |
| Features | Crests and troughs | Compressions and rarefactions |
| Can travel through | Solids, liquid surfaces | Solids, liquids, gases |
| Can be polarised | Yes | No |
| Examples | Light, water waves, rope waves | Sound, slinky compressions |
Similarities
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