Longitudinal Waves
Longitudinal waves
A longitudinal wave is a wave in which the particles of the medium move parallel to the direction of propagation of the wave. In a transverse wave the particles move perpendicular to the direction of propagation; in a longitudinal wave they move back and forth along the same direction as the wave. Sound waves in air are common examples of longitudinal waves.
Introduction and simple investigation
A longitudinal wave can be demonstrated using a slinky spring. Hold one end of the slinky and give the other end a short push and pull along the length of the spring. A disturbance travels down the spring in the same direction as the push. If you tie a ribbon to one coil, the ribbon moves back and forth along the length of the slinky as the disturbance passes; this represents the motion of one particle of the medium.
Repeating the push-pull motion continuously produces a train of pulses that form a longitudinal wave, consisting of alternating regions in which the coils are close together and regions in which they are further apart.
Compression and rarefaction
Instead of crests and troughs (which occur in transverse waves), longitudinal waves have compressions and rarefactions.
Compression - a region in a longitudinal wave where the particles are closest together (higher local pressure).
Rarefaction - a region in a longitudinal wave where the particles are furthest apart (lower local pressure).
Wavelength and amplitude
Wavelength (λ) - the distance between two consecutive points that are in phase. For a longitudinal wave this is the distance between two consecutive compressions or between two consecutive rarefactions.
Amplitude - the maximum displacement from equilibrium. For a longitudinal (pressure) wave, amplitude corresponds to the maximum change in pressure (positive for compressions, negative for rarefactions) or the maximum displacement of particles from their equilibrium positions.
Period and frequency
Period (T) - the time taken by the wave to travel one wavelength or the time between successive occurrences of the same phase at a fixed point. Period is measured in seconds (s).
Frequency (f) - the number of wavelengths (or cycles) that pass a given point per second. Frequency is measured in hertz (Hz).
The relationship between period and frequency is
f = 1/T
or equivalently
T = 1/f
Wave speed
Wave speed (v) is the distance a wave travels per unit time. For a wave that travels one wavelength in one period,
v = distance travelled/time taken = λ/T
Using f = 1/T, this gives the standard wave equation
v = λ · f
where
- v is the wave speed in m · s-1,
- λ is the wavelength in metres (m),
- f is the frequency in hertz (Hz or s-1).
Note: The frequency of a wave is determined by the source of the vibration and does not change when the wave enters a different medium. If the wave speed changes on entering a new medium and the frequency remains the same, the wavelength changes according to v = λ · f.
QUESTION
The musical note "A" is a sound wave. The note has a frequency of 440 Hz and a wavelength of 0.784 m. Calculate the speed of the musical note.
SOLUTION
Determine what is given and what is required.
Given: f = 440 Hz, λ = 0.784 m. Required: v, the wave speed.
Use the wave equation v = f · λ.
v = (440 Hz) × (0.784 m)
v = 344.96 m · s-1
Therefore, the speed of the musical note "A" is approximately 345 m · s-1.
QUESTION
A longitudinal wave travels into a medium in which its speed increases. How does this affect its (write only increases, decreases, stays the same):
- period?
- wavelength?
SOLUTION
The frequency of a longitudinal wave is determined by the source that creates it and therefore is unchanged when the wave enters a different medium. Since the period T = 1/f, the period also remains the same. From v = λ · f, if v increases and f remains the same, the wavelength λ must increase.
Answers:
- period: stays the same
- wavelength: increases
Summary
- A longitudinal wave is a wave in which the particles of the medium move parallel to the direction of wave propagation.
- Longitudinal waves consist of alternating compressions (regions of higher pressure and particle density) and rarefactions (regions of lower pressure and particle density).
- The wavelength of a longitudinal wave is the distance between two consecutive compressions or two consecutive rarefactions.
- The period T and frequency f are related by T = 1/f (or f = 1/T).
- The wave equation is v = λ · f, relating wave speed v, wavelength λ and frequency f.
Physical quantities and units
| Quantity | Unit name | Unit symbol |
|---|---|---|
| Amplitude (A) | metre | m |
| Wavelength (λ) | metre | m |
| Period (T) | second | s |
| Frequency (f) | hertz | Hz (s-1) |
| Wave speed (v) | metre per second | m · s-1 |
End of chapter exercises
1. Which of the following is not a longitudinal wave?
- a. light
- b. sound
- c. ultrasound
2. Which of the following media can a longitudinal wave like sound not travel through?
