Table of Contents
Wave Velocity and Superposition
1. Introduction
A wave is a disturbance that transfers energy from one location to another without the net transfer of matter. Waves occur in many forms, including water waves, sound waves, electromagnetic waves and waves travelling along stretched strings.
Wave velocity describes how quickly a wave disturbance propagates through a medium or through space. The principle of superposition explains how two or more waves interact when they overlap.
These concepts are fundamental to understanding sound, light, communication systems, musical instruments, interference patterns and many other physical phenomena.
2. Wave Velocity
Wave velocity is the distance travelled by a wave disturbance per unit time.
For a periodic wave, its velocity is related to frequency and wavelength by:
[
v=f\lambda
]
Where:
- (v) = wave velocity in metres per second (m/s)
- (f) = frequency in hertz (Hz)
- (\lambda) = wavelength in metres (m)
This equation is one of the most important relationships in wave physics.
2.1 Frequency
Frequency is the number of complete wave cycles passing a fixed point in one second.
[
f=\frac{1}{T}
]
Where (T) is the time period in seconds.
The SI unit of frequency is hertz (Hz).
One hertz represents one complete cycle per second.
2.2 Time Period
The time period is the time required for one complete oscillation or wave cycle.
[
T=\frac{1}{f}
]
A wave with a frequency of 20 Hz has a time period of:
[
T=\frac{1}{20}=0.05\text{ s}
]
2.3 Wavelength
Wavelength is the distance between two successive points in the same phase of a wave.
For example, it is the distance between:
- Two consecutive crests
- Two consecutive troughs
- Two successive compressions in a longitudinal wave
- Two successive rarefactions in a longitudinal wave
The symbol for wavelength is (\lambda).
2.4 Wave Speed and Particle Speed
Wave speed must be distinguished from the speed of individual particles of the medium.
Wave speed describes the propagation of the disturbance. Particle speed describes the motion of particles as they oscillate around their equilibrium positions.
In a travelling wave, energy propagates through the medium while particles generally oscillate about their equilibrium positions.
3. Types of Wave Velocity
3.1 Transverse Waves
In a transverse wave, particles vibrate perpendicular to the direction of wave propagation.
Examples include:
- Waves travelling along a stretched string
- Electromagnetic waves
- Certain seismic S-waves
3.2 Longitudinal Waves
In a longitudinal wave, particles vibrate parallel to the direction of wave propagation.
Examples include:
- Sound waves in air
- Compression waves in a spring
- Seismic P-waves
3.3 Wave Velocity in Different Media
Wave velocity depends on the physical properties of the medium and the type of wave.
For example:
- Sound travels at different speeds in gases, liquids and solids.
- Waves on a string depend on its tension and linear mass density.
- Electromagnetic waves travel at different speeds in different optical media.
For a stretched string:
[
v=\sqrt{\frac{F}{\mu}}
]
Where:
- (F) = tension in the string in newtons
- (\mu) = mass per unit length in kg/m
This shows that increasing tension increases wave speed, whereas increasing linear mass density decreases it.
4. Phase and Phase Difference
Phase describes the position of a particle within its oscillation cycle.
Two points on a wave are in phase when they occupy corresponding positions in their cycles and move in the same direction.
Two points are in opposite phase when their phase difference is (180^\circ) or (\pi) radians.
The phase difference associated with a path difference (\Delta x) is:
[
\Delta\phi=\frac{2\pi\Delta x}{\lambda}
]
Where:
- (\Delta\phi) = phase difference in radians
- (\Delta x) = path difference
- (\lambda) = wavelength
Important phase relationships include:
| Path difference | Phase difference |
|---|---|
| (0) | (0^\circ) |
| (\lambda/4) | (90^\circ) |
| (\lambda/2) | (180^\circ) |
| (3\lambda/4) | (270^\circ) |
| (\lambda) | (360^\circ) |
A path difference equal to one complete wavelength corresponds to a full phase cycle.
5. Principle of Superposition
The principle of superposition states that when two or more waves overlap in a linear medium, the resultant displacement at any point is the algebraic sum of the individual displacements produced by each wave.
Mathematically:
[
y=y_1+y_2+y_3+\cdots
]
Where (y) is the resultant displacement.
The waves do not permanently destroy or alter one another. After overlapping, they continue travelling according to the properties of the medium.
Superposition applies to mechanical waves and electromagnetic waves under appropriate linear conditions.
6. Constructive Interference
Constructive interference occurs when two waves meet in phase, producing a resultant wave of greater amplitude.
For two waves with equal amplitudes, the resultant amplitude is twice the amplitude of either individual wave.
The condition for constructive interference is:
[
\Delta x=n\lambda
]
Where (n=0,1,2,3,\ldots).
For two waves of equal amplitude (A), the resultant amplitude at a phase difference (\Delta\phi) is:
[
A_R=2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|
]
Constructive interference is observed in:
- Bright regions of suitable light interference patterns
- Loud regions produced by sound-wave interference
- Reinforcement of water waves
- Certain radio-wave propagation patterns
7. Destructive Interference
Destructive interference occurs when two waves meet in opposite phase, reducing the resultant amplitude.
For two waves of equal amplitude, complete destructive interference occurs when their phase difference is (180^\circ).
The condition is:
[
\Delta x=\left(n+\frac{1}{2}\right)\lambda
]
For complete destructive interference, the two waves must also have equal amplitudes.
Applications include:
- Active noise cancellation
- Reduction of unwanted vibrations
- Dark regions in light interference patterns
- Sound reduction through carefully designed acoustic systems
Destructive interference does not mean that energy simply disappears. Energy is redistributed through the resulting wave pattern and the surrounding system.
8. Interference of Waves
Interference is the phenomenon in which waves overlap and produce a resultant displacement pattern.
The main types are:
- Constructive interference
- Destructive interference
A stable interference pattern generally requires coherent sources, meaning sources with the same frequency and a constant phase difference.
Interference provides important evidence for the wave nature of light and is also observed in sound and water waves.
9. Standing Waves
A standing wave is formed when two waves of the same frequency, wavelength and amplitude travel in opposite directions and superpose.
Standing waves contain fixed points called nodes and points of maximum displacement called antinodes.
9.1 Nodes
A node is a point where the resultant displacement is always zero.
9.2 Antinodes
An antinode is a point where the resultant displacement has maximum amplitude.
The distance between two consecutive nodes is:
[
\frac{\lambda}{2}
]
The distance between a node and its nearest antinode is:
[
\frac{\lambda}{4}
]
Standing waves are important in musical instruments, resonance systems and the study of vibrating strings.
10. Reflection and Transmission of Waves
When a wave reaches a boundary between two media, several outcomes are possible:
- Reflection: the wave returns into the original medium.
- Transmission: the wave enters the second medium.
- Absorption: some wave energy is absorbed by the material.
- Refraction: the direction and speed of the wave change when it enters another medium at an angle.
During reflection, the frequency remains unchanged because it is determined by the source. The wavelength may change if the wave enters a medium where its speed differs.
11. Worked Numerical Examples
Example 1: Calculating Wave Velocity
A wave has a frequency of 50 Hz and a wavelength of 4 m. Calculate its velocity.
Given:
[
f=50\text{ Hz},\quad \lambda=4\text{ m}
]
Using:
[
v=f\lambda
]
[
v=50\times4=200\text{ m/s}
]
Answer: The wave velocity is 200 m/s.
Example 2: Calculating Wavelength
A sound wave travels at 340 m/s and has a frequency of 170 Hz. Find its wavelength.
[
\lambda=\frac{v}{f}
]
[
\lambda=\frac{340}{170}=2\text{ m}
]
Answer: The wavelength is 2 m.
Example 3: Calculating Frequency
A wave travels at 300 m/s and has a wavelength of 5 m. Calculate its frequency.
[
f=\frac{v}{\lambda}
]
[
f=\frac{300}{5}=60\text{ Hz}
]
Answer: The frequency is 60 Hz.
Example 4: Wave Speed on a String
A string has a tension of 100 N and a linear mass density of 0.01 kg/m. Calculate the wave speed.
[
v=\sqrt{\frac{F}{\mu}}
]
[
v=\sqrt{\frac{100}{0.01}}
]
[
v=\sqrt{10000}=100\text{ m/s}
]
Answer: The wave speed is 100 m/s.
Example 5: Phase Difference
Two points on a wave are separated by a distance equal to one-quarter of its wavelength. Calculate their phase difference.
[
\Delta\phi=\frac{2\pi\Delta x}{\lambda}
]
Since (\Delta x=\lambda/4):
[
\Delta\phi=\frac{2\pi}{4}=\frac{\pi}{2}
]
Answer: The phase difference is (90^\circ) or (\pi/2) radians.
12. Applications of Wave Velocity and Superposition
Wave velocity and superposition are used in numerous scientific and technological applications:
- Telecommunications: understanding the propagation and interference of radio waves.
- Acoustics: designing auditoriums and controlling sound distribution.
- Medical imaging: using ultrasound waves and their reflections.
- Seismology: studying earthquake waves and the internal structure of Earth.
- Noise cancellation: producing sound waves that reduce unwanted noise.
- Musical instruments: producing standing waves in strings and air columns.
- Optical instruments: using interference in precision measurement.
- Radar systems: analysing reflected electromagnetic waves.
- Engineering: studying vibration and wave behaviour in structures.
- Oceanography: examining the movement and interaction of water waves.
13. Common Examination Distinctions
- Wave velocity is not the same as particle velocity.
- Frequency is determined by the source and remains unchanged when a wave enters another medium.
- Wavelength changes when wave speed changes while frequency remains constant.
- Constructive interference increases resultant amplitude.
- Destructive interference reduces resultant amplitude.
- Complete destructive interference requires equal amplitudes and opposite phases.
- Nodes have zero resultant displacement in an ideal standing wave.
- Antinodes have maximum resultant displacement.
- A standing wave does not transport energy along the medium in the same way as a progressive travelling wave.
- Superposition is the basis of interference.
36 Key Takeaways for Exams
- A wave transfers energy without net transfer of matter.
- Wave velocity is the speed at which a wave disturbance propagates.
- The SI unit of wave velocity is metre per second.
- The fundamental wave equation is (v=f\lambda).
- Frequency is the number of complete cycles per second.
- The SI unit of frequency is hertz.
- Time period is the time required for one complete cycle.
- Frequency and time period are reciprocals.
- Wavelength is the distance between successive points in the same phase.
- Wave velocity differs from the oscillatory velocity of particles.
- Transverse-wave particles vibrate perpendicular to propagation.
- Longitudinal-wave particles vibrate parallel to propagation.
- Wave speed depends on the properties of the medium and the type of wave.
- The speed of a wave on a stretched string depends on tension and linear mass density.
- Increasing string tension increases wave speed.
- Increasing linear mass density decreases wave speed.
- Phase describes the position of a particle within its oscillation cycle.
- A phase difference of (180^\circ) represents opposite phase.
- A path difference of one wavelength corresponds to a phase difference of (360^\circ).
- The principle of superposition states that overlapping displacements add algebraically.
- Constructive interference increases resultant amplitude.
- Destructive interference decreases resultant amplitude.
- Complete destructive interference requires equal amplitudes and opposite phases.
- Coherent sources maintain a constant phase difference.
- Interference occurs in sound, light and water waves.
- Standing waves form through the superposition of oppositely travelling waves.
- Nodes are points of zero resultant displacement.
- Antinodes are points of maximum displacement.
- The distance between consecutive nodes is half a wavelength.
- The distance between a node and its nearest antinode is one-quarter wavelength.
- Frequency remains unchanged when a wave enters a new medium.
- Wavelength changes when wave speed changes at constant frequency.
- Reflection returns a wave into its original medium.
- Transmission allows a wave to enter another medium.
- Active noise cancellation uses destructive interference.
- Musical instruments commonly use standing waves to produce sound.