Table of Contents
Wave Motion, Frequency, Wavelength and Amplitude
1. Introduction
Wave motion is the propagation of a disturbance through a medium or space, resulting in the transfer of energy from one location to another.
Waves are characterised by several measurable quantities, including frequency, wavelength, amplitude, time period, phase and speed.
Understanding these quantities is essential for studying sound, light, communication systems, ocean waves, seismic activity and other physical phenomena.
2. Wave Motion
2.1 Definition
Wave motion is the process by which a disturbance travels through a medium or space, transferring energy without requiring the bulk transfer of matter.
In mechanical waves, particles of the medium oscillate about their equilibrium positions. The disturbance moves forward while individual particles generally remain near their original positions.
2.2 Particle Motion and Wave Propagation
The direction in which particles oscillate depends on the type of wave.
- In transverse waves, particle oscillation is perpendicular to the direction of propagation.
- In longitudinal waves, particle oscillation is parallel to the direction of propagation.
For example, in a wave travelling along a stretched horizontal string, the string particles may move vertically while the disturbance travels horizontally.
In a sound wave, air particles oscillate back and forth parallel to the direction in which sound travels.
3. Important Quantities of Wave Motion
3.1 Amplitude
Amplitude is the maximum displacement of an oscillating particle from its equilibrium position.
It is represented by (A) or (a), depending on convention.
Its SI unit is the metre for mechanical displacement.
Amplitude determines the maximum extent of oscillation. For many linear waves, energy transported is proportional to the square of amplitude, provided other relevant conditions remain unchanged.
For example, doubling the amplitude of a sinusoidal wave can produce four times the average energy intensity under comparable conditions.
3.2 Wavelength
Wavelength is the distance between two successive points that are in the same phase of oscillation.
It is represented by the Greek letter lambda, (\lambda).
Its SI unit is the metre (m).
Examples include:
- Distance between two successive crests.
- Distance between two successive troughs.
- Distance between two successive compressions.
- Distance between two successive rarefactions.
A shorter wavelength means that more wave cycles occupy a given distance.
3.3 Frequency
Frequency is the number of complete oscillations or cycles produced per second.
It is represented by (f) or (\nu).
Its SI unit is the hertz (Hz).
[
1\ Hz=1\ cycle\ per\ second
]
A wave with a frequency of 50 Hz completes 50 cycles every second.
3.4 Time Period
The time period is the time taken to complete one full oscillation or cycle.
It is represented by (T).
Its SI unit is the second (s).
The relationship between frequency and time period is:
[
f=\frac{1}{T}
]
and:
[
T=\frac{1}{f}
]
A high-frequency wave has a short time period, while a low-frequency wave has a long time period.
3.5 Wave Speed
Wave speed is the distance travelled by a wave disturbance per unit time.
It is represented by (v).
Its SI unit is metres per second (m s(^{-1})).
For a periodic wave, speed depends on frequency and wavelength:
[
v=f\lambda
]
This relationship is fundamental to wave physics.
3.6 Angular Frequency
Angular frequency measures the rate of change of phase in radians per second.
It is represented by (\omega).
[
\omega=2\pi f
]
Its SI unit is radian per second (rad s(^{-1})).
3.7 Wave Number
Wave number describes the spatial rate of phase change.
It is represented by (k).
[
k=\frac{2\pi}{\lambda}
]
Its SI unit is radian per metre (rad m(^{-1})).
4. Relationship Between Frequency, Wavelength and Speed
The fundamental wave-speed relationship is:
[
v=f\lambda
]
Where:
- (v) = wave speed.
- (f) = frequency.
- (\lambda) = wavelength.
The equation shows that wave speed is the product of frequency and wavelength.
For a wave travelling through a particular medium under unchanged conditions, its speed is generally determined by the properties of that medium.
If the frequency increases while wave speed remains constant, wavelength decreases.
If frequency decreases while wave speed remains constant, wavelength increases.
5. Factors Affecting Wave Speed
Wave speed depends on the nature of the wave and the physical properties of the medium.
5.1 Wave Speed in a Stretched String
The speed of a transverse wave on an ideal stretched string is:
[
v=\sqrt{\frac{F}{\mu}}
]
Where:
- (F) = tension in the string in newtons.
- (\mu) = mass per unit length in kg m(^{-1}).
Increasing tension increases wave speed, while increasing mass per unit length decreases wave speed.
5.2 Sound Speed in a Medium
The speed of sound in a fluid depends on its bulk modulus and density.
[
v=\sqrt{\frac{B}{\rho}}
]
Where:
- (B) = bulk modulus.
- (\rho) = density of the medium.
For an ideal gas:
[
v=\sqrt{\frac{\gamma RT}{M}}
]
Where:
- (\gamma) = ratio of specific heat capacities.
- (R) = universal gas constant.
- (T) = absolute temperature.
- (M) = molar mass.
The speed of sound in air generally increases with temperature.
5.3 Speed of Light
In a vacuum, electromagnetic waves travel at approximately:
[
c=3.0\times10^8\ m,s^{-1}
]
In a transparent material, the phase velocity of light is generally lower than its vacuum speed.
The refractive index is:
[
n=\frac{c}{v}
]
Where (v) is the phase velocity of light in the material.
6. Wave Equation
A sinusoidal travelling wave moving in the positive x-direction can be represented by:
[
y(x,t)=A\sin(kx-\omega t+\phi)
]
Where:
- (y) = displacement.
- (A) = amplitude.
- (k) = wave number.
- (x) = position.
- (\omega) = angular frequency.
- (t) = time.
- (\phi) = initial phase constant.
For a wave travelling in the negative x-direction, the sign between the spatial and time-dependent terms is reversed.
The equation describes how the displacement of a particle varies with both position and time.
7. Phase and Phase Difference
7.1 Phase
Phase describes the position of an oscillating particle within its cycle.
It determines whether a particle is at maximum displacement, minimum displacement or an intermediate position.
7.2 Phase Difference
Phase difference is the difference in phase between two oscillations or two points on a wave.
It may be expressed in radians or degrees.
One complete cycle corresponds to:
[
2\pi\ radians=360^\circ
]
Two points separated by one complete wavelength have the same phase.
Two points separated by half a wavelength have a phase difference of (\pi) radians, or 180°.
8. Energy and Intensity of Waves
Waves transfer energy from one location to another.
8.1 Wave Energy
For many linear mechanical waves, the energy transported is proportional to the square of the amplitude.
[
E\propto A^2
]
The exact expression depends on the wave type and the properties of the medium.
8.2 Wave Intensity
Intensity is the power transmitted per unit area perpendicular to the direction of propagation.
[
I=\frac{P}{A_s}
]
Where:
- (I) = intensity in W m(^{-2}).
- (P) = power in watts.
- (A_s) = area in square metres.
For an ideal point source radiating uniformly in all directions:
[
I=\frac{P}{4\pi r^2}
]
Where (r) is the distance from the source.
This shows that intensity decreases with the square of distance in an ideal free-field situation.
9. Frequency and Pitch of Sound
Frequency is a major factor determining the perceived pitch of a sound.
- Higher frequency generally produces higher pitch.
- Lower frequency generally produces lower pitch.
The human hearing range is commonly approximated as 20 Hz to 20,000 Hz for young people with healthy hearing, although it varies among individuals and declines with age.
9.1 Infrasound
Infrasound refers to sound waves with frequencies below approximately 20 Hz.
Sources include earthquakes, volcanic activity, large machinery and some animal communication systems.
9.2 Audible Sound
Audible sound generally falls within the approximate frequency range of 20 Hz to 20 kHz.
9.3 Ultrasound
Ultrasound refers to sound waves with frequencies above approximately 20 kHz.
Applications include:
- Medical imaging.
- Industrial testing.
- Ultrasonic cleaning.
- Distance measurement.
- Sonar systems.
10. Doppler Effect
The Doppler effect is the apparent change in frequency of a wave due to relative motion between the source and the observer.
When a sound source approaches an observer, the observed frequency generally increases.
When it moves away, the observed frequency generally decreases.
Applications include:
- Measuring vehicle speed.
- Medical Doppler ultrasound.
- Weather radar.
- Astronomy.
- Satellite tracking.
The Doppler effect applies to sound and electromagnetic waves, although their equations differ according to the physical circumstances.
11. Wave Speed in Different Media
Different waves behave differently in different materials.
11.1 Sound Waves
Sound generally travels faster in solids than in gases because of the relationship between stiffness and density.
The actual speed depends on the material and wave type.
11.2 Light Waves
Light travels fastest in a vacuum. Its speed in a material depends on the material’s refractive index and the wavelength.
11.3 Water Waves
Water-wave speed depends on factors such as water depth, wavelength and surface tension.
In deep-water gravity waves, longer wavelengths generally travel faster than shorter wavelengths.
12. Worked Numerical Examples
Example 1: Calculating Wave Speed
A wave has a frequency of 50 Hz and a wavelength of 4 m. Calculate its speed.
Given:
[
f=50\ Hz,\quad \lambda=4\ m
]
Formula:
[
v=f\lambda
]
Solution:
[
v=50\times4
]
[
v=200\ m,s^{-1}
]
Answer: The wave speed is 200 m s(^{-1}).
Example 2: Calculating Frequency
A wave travels at 300 m s(^{-1}) and has a wavelength of 6 m. Calculate its frequency.
[
f=\frac{v}{\lambda}
]
[
f=\frac{300}{6}=50\ Hz
]
Answer: The frequency is 50 Hz.
Example 3: Calculating Wavelength
A sound wave travels at 340 m s(^{-1}) and has a frequency of 170 Hz. Calculate its wavelength.
[
\lambda=\frac{v}{f}
]
[
\lambda=\frac{340}{170}=2\ m
]
Answer: The wavelength is 2 m.
Example 4: Calculating Time Period
A vibrating object completes 25 oscillations per second. Calculate its time period.
[
T=\frac{1}{f}
]
[
T=\frac{1}{25}=0.04\ s
]
Answer: The time period is 0.04 seconds.
Example 5: Wave Speed on a String
A string has a tension of 100 N and a mass per unit length of 0.01 kg m(^{-1}). Calculate the wave speed.
[
v=\sqrt{\frac{F}{\mu}}
]
[
v=\sqrt{\frac{100}{0.01}}
]
[
v=\sqrt{10000}=100\ m,s^{-1}
]
Answer: The wave speed is 100 m s(^{-1}).
Example 6: Intensity of a Point Source
A point source radiates 120 W uniformly in all directions. Calculate the intensity at a distance of 2 m.
[
I=\frac{P}{4\pi r^2}
]
[
I=\frac{120}{4\pi(2)^2}
]
[
I=\frac{120}{16\pi}
]
[
I\approx2.39\ W,m^{-2}
]
Answer: The intensity is approximately 2.39 W m(^{-2}).
13. Common Examination Errors
- Confusing frequency with wavelength.
- Treating amplitude as the distance between two crests.
- Using centimetres instead of metres without conversion.
- Confusing frequency with time period.
- Assuming wave speed always changes when frequency changes.
- Forgetting that wave speed depends on the medium.
- Using degrees and radians interchangeably without conversion.
- Confusing wave intensity with total wave power.
- Assuming sound can travel through a vacuum.
- Confusing pitch with loudness.
36 Key Takeaways for Exams
- Wave motion transfers energy through the propagation of a disturbance.
- Mechanical waves require a material medium.
- Electromagnetic waves can travel through a vacuum.
- Amplitude is the maximum displacement from equilibrium.
- Wavelength is the distance between successive points in the same phase.
- Frequency is the number of complete cycles per second.
- Frequency is measured in hertz.
- Time period is the time required for one complete cycle.
- Frequency and time period are reciprocals.
- Wave speed is measured in metres per second.
- Wave speed is the product of frequency and wavelength.
- Frequency and wavelength are inversely related when wave speed remains constant.
- Angular frequency is measured in radians per second.
- Angular frequency equals (2\pi f).
- Wave number equals (2\pi/\lambda).
- A transverse wave has perpendicular particle oscillations.
- A longitudinal wave has parallel particle oscillations.
- Wave speed on an ideal string depends on tension and mass per unit length.
- Greater string tension increases wave speed.
- Greater mass per unit length decreases wave speed.
- Sound speed in a fluid depends on bulk modulus and density.
- Sound generally travels faster in solids than in gases.
- Light travels at approximately (3.0\times10^8\ m,s^{-1}) in vacuum.
- Refractive index is the ratio of light speed in vacuum to phase velocity in a medium.
- Phase describes the position of an oscillation within its cycle.
- One complete cycle corresponds to (2\pi) radians or 360°.
- Wave energy is proportional to the square of amplitude for many linear waves.
- Intensity is power transmitted per unit area.
- Ideal point-source intensity decreases with the square of distance.
- Higher sound frequency generally produces higher pitch.
- Infrasound has frequencies below approximately 20 Hz.
- Ultrasound has frequencies above approximately 20 kHz.
- The Doppler effect is an apparent frequency change caused by relative motion.
- Medical Doppler ultrasound measures motion, including blood flow.
- Wave speed depends on the properties of the medium and the type of wave.
- Frequency, wavelength, amplitude and speed are fundamental quantities in wave physics.