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Physics for Civil Engineering

This is an introduction to Electricity, Strength of Materials and Waves.

Lecture 14 (Waves, Wave Equation and Intensity)

In this lecture the following are introduced:
• Wave motion as an energy transfer
• Types of waves
• Basic Basic Wave Parameters
• Representing Moving Shapes
• Transverse Sinusoidal Waves
• The Intensity, Impedance and Pressure Amplitude of a Wave
• Intensity Level and decibel scale
• Hearing Loss
• The Fletcher-Munson Curves and the Phon
• Pitch

Waves transfer energy

Any vibrating body that is connected to its environment will transfer energy to its environment. The vibrations are then transferred though the environment from neighbour to neighbour. This energy transfer is called wave motion. Waves move energy through a medium without moving the whole medium.

Leonardo di Vinci
"waves made in a field of grain by the wind, ... we see the waves running across the field while the grain remains in place."

Types of waves

Longitudinal waves

When waves transfer energy by pushing neighbours in the same direction that the energy moves, the waves are called longitudinal waves. In the simulation below (from Dr. Daniel A. Russell) you can see energy move to the right while individual particles vibrate to the left and right about fixed points.

longitudinal wave

The places when the particles cluster together are volumes of high pressure so these waves are also called pressure waves. Sound waves are an example of pressure waves and they can move through gases, liquids and solids. For sound waves, the denser the medium the faster the speed.
Speed through air (1atm, 200) =344 m.s-1
Speed through sea water = 1531 m.s-1
Speed through iron = 5130 m.s-1

Transverse waves

When waves transfer energy by pulling neighbours sideways to the direction of travel, the waves are called transverse waves. In the simulation below (also from Dr. Daniel A. Russell) you can see energy move to the right while individual particles vibrate up and down about fixed points.

transverse wave

Electromagnetic waves (X-rays, light, radio, radar and TV waves) are examples of transverse waves formed by electric and magnetic fields vibrating together at right angles to the wave's motion. They don't need any medium so they can move through a vacuum, (good for us or we wouldn't see the Sun!). They all move at the same speed of 300,000 km.s-1 when they travel through vacuum. They slow down when they travel through a medium (this is an average speed between interactions).

Mechanically twisting or pulling a medium sideways is called shearing so waves formed this way are also called shear waves.

Longitudinal and Transverse waves together

Sometimes longitudinal and transverse waves occur together. Ocean waves are a combination of longitudinal and transverse waves because the surface of the water can be pulled sideways as well as pushed longitudinally. In the simulation below (also from Dr. Daniel A. Russell) you can see energy move to the right while individual particles move clockwise in circles or ellipses.

water wave

When ocean waves get to a shelving beach the speed of the waves changes relative to each other and circles go to ellipses and then the wave breaks.

Seismic waves are formed when there is a sudden movement (or slip) between layers in the Earth's crust. This may happen anywhere between several km and several 100s km down from the surface. The wave motions that occur through the crust have Pressure ("P") components and Shear ("S") components.
The P waves move at 5 - 14 km. s-1
The S waves move at 3 - 8 km. s-1
When they reach the surface an Earthquake occurs, and the timing between the arrivals of the The S and P waves and their sizes at different places will enable the epicentre to be determined.

Earth cross-section

(Note: seismic waves can also have "surface" waves. Again see Dr. Daniel A. Russell's excellent pages for details.)

Basic Wave Parameters

The amplitude A, is half the height difference between a peak and a trough.
The wavelength λ, is the distance between successive peaks (or troughs).
The period T, is the time between successive peaks (or troughs).
The wave speed c, is the speed at which peaks (or troughs) move.
The frequency ν, (Greek letter "nu") measures the number of peaks (or troughs) that pass per second.
Note that "c" (from the Latin word "celeritas" meaning swiftness) is used for wave speed, not "v". One reason is so that it is not confused with frequency.

wave diagram

wave speed equation

Example W1
Seismic Shear waves travel at 4000 m.s-1 and they have a period of 0.12s. Find the wavelength of these waves.

Answer W1

solution W1

Representing Moving Shapes

The station below has a frame of reference with axes labelled x and y.
The engine and carriage below have a frame of reference at the end of the first carriage.
It has axes labelled X and Y.
The shape Y = F(X) is drawn on the side of the carriage in the (X,Y) frame of reference.
The engine and carriage are moving at a constant speed c to the right (positive x axis).

At time t = 0 s, the carriage and the station axes coincide.

train at station

At time t s later:

train moving past station

Vertical distance references do not change, i.e. Y = y at all times.
The horizontal distance between the Y and y axes increases uniformly with time, i.e. d = ct.

The distance X from the carriage origin to a point P on the side of the carriage will not change in time.
The distance x from the station origin to the point P will increase with time, i.e. x = X + ct

This means that the reference frames transform as:

(x,y) ⇒ (X + ct, Y) and
(X,Y) ⇒ (x - ct, y)

In the carriage frame of reference: Y = F(X) defines a shape on the side of the carriage.
In the station frame of reference: y = f(x - ct) defines the same shape f(x) moving at a speed c to the right.

Representing Transverse Sinusoidal Waves

You can only have the sine of an angle. To represent a sine shape in space, the x distance has to be converted to get an angle.

One complete cycle in space (one wavelength) is equivalent to one cycle in phase ( radian).
A fractional distance of a wavelength will equal the same fraction of .

same fractions
sine curve with different reference axes

φ is a phase angle, it relates a distance in space to a fraction of a wavelength.

The relationship between phase angle and distance is given by:

phase angle and distance

k is called the wave number:

wave number

It is measured in radians per metre (rad.m-1).

The wave speed is:

wave speed

It is measured in metres per second (m.s-1).

ω (above right), is the angular frequency:

angular frequency definition

It is measured in radians per second (rad.s-1).

ν is the linear frequency:

angular frequency definition

It is measured in Hertz (s-1).

A sine shape in space is given by:

sine shape in space

A sine wave moving to the right (positive x direction) at speed c will be written:

sine wave moving to the right

A sine wave moving to the left (negative x direction) at speed c will be written:

sine wave moving to the left

The time part of the wave can be written:

wave number and angular frequency

A sine wave moving to the right is:

sine wave moving to the right

Also written as:

wave equation in k and omega

Adding an initial phase α, (Greek letter "alpha") tells what is happening at time 0 s.

wave equation with initial phase

Particle Motion in Transverse Sinusoidal Waves

transverse wave

From Dr. Daniel A. Russell's transverse wave diagram above, while the wave moves at constant speed in the +x direction (to the right), the vibrating particles which make up the wave, move with simple harmonic motions in the y direction .
In concentrating on the particles, it is seen that neighbouring particles have slightly different x values which appear as slightly different initial phases in a Simple Harmonic Motion.
At position x1, the equation y = A sin (kx1 - ωt + α) is effectively y = A sin (α1 - ωt), i.e. an SHM.

The transverse displacements of particles are governed by:

wave equation with initial phase

The transverse particle speeds are given by:

transverse speed

Notice that displacement and particle speed are 90° out of phase (sines and cosines). When the particle is at its largest displacement, there is zero particle velocity. Maximum transverse particle velocity occurs as the particle crosses the axis.

Example W2
A sinusoidal wave has a wavelength of 1.4m. Find the phase difference between a point 0.3m from the peak of a wave and another point 0.7m further along from the same peak.

Answer W2

solution to W2

Example W3
The equation of a transverse sinusoidal wave is given by: equation for W3.
(a) the amplitude of the wave,
(b) the wavelength,
(c) the frequency,
(d) the wave speed, and
(e) the displacement at position 0 m and time 0 s.
(f) the maximum transverse particle speed.

Answer W3

Amplitude, A is 2 mm.

solution W3 b

solution W3 c

solution W3 d

solution W3 e

solution W3 f

The Intensity, Impedance and Pressure Amplitude of a Wave

In general, an Intensity is a ratio. For example, pressure is the intensity of force as it is force/area. Also, density (symbol ρ) is the intensity of mass as it is mass/volume.
The Intensity of waves (called Irradiance in Optics) is defined as the power delivered per unit area.
The unit of Intensity will be W.m-2.

definition of intensity

The wave energy comes from the simple harmonic motion of its particles. The total energy will equal the maximum kinetic energy.

energy equation

Combining these two results:

Intensity equation in symbols

The Impedance of the medium (called the Specific Acoustic Impedance in Acoustics) is defined by the product of density and wave speed.
In symbols: Impedance, z = ρc with a unit of Pa.s.m-1.

The quantity is the maximum transverse speed of the particles, so it has m.s-1.

It can be seen that the intensity of a wave increases with its wavespeed c, amplitude A, and frequency ω.

Multiplying top and bottom by ρc:

Intensity with Pressure amplitude

P0 is called the pressure amplitude, because when the unit for Impedance (Pa.s.m-1) is combined with , a transverse speed term (m.s-1), it has the unit of Pressure (Pascal). It is useful when dealing withpressure waves.

Example W4
A wave of frequency 1000 Hz travels in air of density 1.2 kg.m-3 at 340 m.s-1. If the wave has intensity 10 μW.m-2, find the displacement and pressure amplitudes.

Answer W4

solution for W4

Intensity Level

The intensity of a sound is given by power/area. It is an objective measurement and has the unit of W.m-2. Loudness is a subjective perception. For a long time it was thought that the ear responded logarithmically to sound intensity, i.e. that an increase of 100× in intensity (W.m-2) would be perceived as a loudness increase of 20×.

The Intensity Level was defined to represent loudness. It was accordingly based on a logarithmic scale and has the unit of Bel (after Alexander Graham Bell, not the Babylonian deity).

The deciBel (β) is commonly used as the smallest difference in loudness that can be detected.

definition of decibel

The reference intensity I0 = 10-12 W.m-2 is the (alleged) quietest sound that can be heard. Only about 10% of people can hear this 0 dB sound and that only in the frequency range of 2kHz to 4kHz. About 50% of people can hear 20dB at 1kHz. (The frequency response will be looked at later.)

Approximate Intensity Levels

Type of sound

Intensity level at ear (dB)

Threshold of hearing


Rustle of leaves


Very quiet room


Average room




Busy street


Loud radio


Train through station




Threshold of discomfort


threshold of pain


damage to ear drum


Example W5
The average intensity level for each of two radios is set to 45dB. They are tuned to different radio stations. Find the average intensity level when they are both turned on.

Answer W5

solution to W%

Here the Intensity doubles but the Intensity Level goes up by only 0.3 dB.

Example W6
Sound radiates in a hemi-sphere from a rock band. If the sound level is 100 dB at 10 m, then find the sound level at 4 m.

Answer W6

First find the Intensity from the Intensity Level.

Intensity for answer W6

Calculate the new Intensity. The key is that the radiated power is fixed and as it spreads out over greater areas the Intensity decreases in accordance with the following relationship.

new intensity calculation

Calculate the new Intensity Level.

new intensity level calculation

Other Loudness measures

There are other ways of representing the human response, some of these are:

definition of loudness

This puts the threshold of hearing at 4dB.


definition of Sones

1 Sone = 40dB at 1kHz.

Degrees of Hearing Loss

Hearing loss information.

A person can have up to 25 dB hearing level and still have "normal" hearing. Those with a mild hearing loss (26-45 dB) may have difficulty hearing and understanding someone who is speaking from a distance or who has a soft voice. They will generally hear one-on-one conversations if they can see the speaker's face and are close to the speaker. Understanding conversations in noisy backgrounds may be difficult. Those with moderate hearing loss (46-65 dB) have difficulty understanding conversational levels of speech, even in quiet backgrounds. Trying to hear in noisy backgrounds is extremely difficult. Those with severe hearing loss (66-85 dB) have difficulty hearing in all situations. Speech may be heard only if the speaker is talking loudly or at close range. Those with profound hearing loss (greater than 85 dB HL) may not hear even loud speech or environmental sounds. They may not use hearing as a primary method of communicating.

The Fletcher-Munson Curves

Fletcher and Munson were researchers who first accurately measured and published a set of curves showing the human's ear's frequency sensitivity versus loudness.  The curves show the ear to be most sensitive to sounds in the 3 kHz to 4 kHz area, a range that corresponds to ear canal resonances.

The lines give a unit called the phon. 100Hz at 71dB has the same apparent loudness as 60dB at 1kHz and hence it is 60 phons. The important range for speech is 300Hz - 3000Hz. Loud noise and age cause the high frequency response to decline.

D.W. Robinson and R.S. Dadson, re-did these lines in 1956 in an article titled: 'A re-determination of the equal-loudness relations for pure tones', British Journal of Applied Physics, 7, 1956, 166-181. These data are generally regarded as being more accurate than those of Fletcher and Munson.

Both sources apply only to pure tones in otherwise silent free-field conditions, with a frontal plane wave etc.


Frequency is measured objectively in Hertz. The subjective sensation of frequency is called the pitch of the note. The ear is not linear with frequency (Hertz). There is a "S" shaped curve between frequency and pitch. The ear is reasonably OK in the range 400Hz to 2.4kHz, but outside this range perception is pitch and frequency differ. For example, 300Hz is perceived as 500Hz, but 10kHz is perceived to be 3kHz. The subjective determination of frequency has a unit called the mel, and is thought to be due to the variable elastic properties of the basilar membrane in the ear.


Waves move energy through a medium without moving the whole medium.
In longitudinal waves the vibration is in the same orientation as the wave movement.
In transverse waves the vibration is at right angles to the wave movement.
Amplitude: A, is half the wave height.
Wavelength: λ, is the distance between successive maxima (or minima).
Frequency: ν, is the number of maxima (or minima) that pass per second and the reciprocal of the Period, T.

Wavespeed:   Wavespeed

Angular frequency:   Angular frequency

Wave Number:   Wave Number

Phase Angle:   Phase Angle

Sinusoidal wave:   Sinusoidal wave

Intensity:   Intensity

Pressure Amplitude:   Pressure Amplitude

deciBel:   deciBel

Hearing depends on both frequency and intensity.

Frequency is measured physically in Hertz and subjectively in Mels.

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