Vibrations and Waves

Periodic Motion

• Motion that repeats themselves over and over again at equal intervals of time.

Hooke’s Law

• Fs = - k x

• Fs is the spring force

• k is the spring constant• It is a measure of the stiffness of the spring

• A large k indicates a stiff spring and a small k indicates a soft spring

• x is the displacement of the object from its equilibrium position• x = 0 at the equilibrium position

• The negative sign indicates that the force is always directed opposite to the displacement

Simple Harmonic Motion

• Motion that occurs when the net force along the direction of motion obeys Hooke’s Law• The force is proportional to the displacement and always

directed toward the equilibrium position

• The motion of a spring mass system is an example of Simple Harmonic Motion

Spring- Mass System

When a body is displaced fromits equilibrium position, thespring force tends to restore itto equilibrium. (Restoring Force)

Amplitude, Period, Frequency and Angular Frequency

• Amplitude - denoted by A, is the maximum magnitude of displacement from equilibrium. (SI unit is in m)

• Period – is the time to complete one cycle – say from A to –A and back to A. (SI unit is in seconds)

• Frequency – is the number of cycles per unit time. (SI unit is in hertz)

• Angular Frequency – denoted by ω, is equal to 2𝜋𝑓. It represents the rate of change of an angular quantity

Waves

• We encounter waves in many situations• Speech and hearing rely on wave propagation. • Modern telecommunications networks such as mobile

phones rely on waves.• Many key areas of Physics, Mathematics and Chemistry

are best described by waves and their interactions.• The way atoms bind together to form molecules can be

understood by the overlap of waves.

Types of waves

• There are several different types of wave that we must consider.

• Mechanical Waves:- These need a medium to propagate in - sound waves.

• Non-mechanical waves:-These waves do not need a medium in which to propagate - light waves.

• Matter waves:- Particles such as protons and electrons can betreated as waves. This forms the basis of quantum mechanics. We willnot be discussing this type of wave in this course.

Examples of Mechanical Waves

• Earthquake / seismic waves

• Water waves

• Sound waves

• Waves that travel down a spring or rope

Nature of Waves

1. A wave is a traveling disturbance.

2. A wave carries energy from place to place.

Longitudinal Wave

Nature of Waves

Transverse Wave

Nature of Waves

TRANSVERSE

WAVE

LONGITUDINAL

WAVE

WATER WAVE

(Long + Trans

Combined)

Periodic Waves

Periodic waves consist of cycles or patterns that are produced over and

over again by the source.

In the figures, every segment of the slinky vibrates in simple harmonic

motion, provided the end of the slinky is moved in simple harmonic

motion.

In the drawing, one cycle is shaded in color.

The period is the time required for one complete cycle.

The frequency is related to the period and has units of Hz, or s-1.

The angular frequency is the rate of change of an angular quantity.

Tf

1

Periodic Waves

T=1

𝑓= 2𝜋/𝜔

𝜔 = 2𝜋f

Graphing Wave Functions

33

Amplitude A is

the maximum

excursion of a

particle of the

medium from

the particle’s

undisturbed

position.

Wavelength

is the horizontal

length of one

cycle of the

wave.

Crest

Trough

Example

The distance between the crest of a water wave and the

next trough is 2 m. If the frequency of a particular wave is 2

Hz, what is the speed of the wave?

(a) 4 m/s

(b) 1 m/s

(c) 8 m/s

(d) 2 m/s

(e) impossible to determine from the information given

Find the amplitude, wavelength, speed and period of the wave if

it has a frequency of 8.00 Hz. Δx = 40.0 cm, Δy = 15.0 cm

A wave with a frequency of 12.3 Hz is traveling from left to right

across a rope as shown in the diagram at the right Positions A and B

in the diagram are separated by a horizontal distance of 42.8 cm.

Positions C and D in the diagram are separated by a vertical distance

of 12.4 cm. Determine the amplitude, wavelength, period and speed of

this wave.

• During wave motion, the particle

are displaced some distance y in

the direction perpendicular to the

x-axis.

• The motion of the particle on the

right lags behind the motion of the

particle on the left by an amount of

time proportional to the distance

between the particles.

Example:

Particle Velocity and Acceleration in a

Sinusoidal Wave

• A disturbance can propagate as a wave along the x-

axis with wave speed v.

• Electric and Magnetic fields satisfy the wave equation,

the wave speed turns out to be the speed of light.

• At which time is point A on the

string moving upward with

maximum speed?

• At which time does point B on

the string have the greatest

upward acceleration?

• At which time does point C on

the string have a downward

acceleration but an upward

velocity?

Energy in Wave Motion

• Power is the instantaneous rate at which energy is

transferred along the string. (It depends on x and t)

• For a sinusoidal wave (sine function), the power is either

positive or zero.

• Energy is transferred to the direction of wave

propagation.

• What happens to power if frequency is doubled for the

same amplitude?

– Power is quadrupled.

Ans. A = 4.51 x10-3

Wave Intensity

• The time average rate at which energy is transported by

the wave per unit area, across a surface perpendicular to

the direction of propagation.

• SI Unit : W/m2 (watts per square meter)

Wave Reflection

The boundary conditions play a role in wave reflections.

Interference of Waves

• Two traveling waves can meet and pass through each

other without being destroyed or even altered

• Waves obey the Superposition Principle

– When two or more traveling waves encounter each other while

moving through a medium, the resulting wave is found by adding

together the displacements of the individual waves point by point

– Actually only true for waves with small amplitudes

Standing Waves on a String

• It is called a standing wave

because the wave pattern

remains in the same position

along the string and its

amplitude fluctuates.

• Nodes – points that never

move at all

• Antinodes – points where the

amplitude of motion is

greatest

• The picture on the left is a

travelling wave. The wave

does move along the string,

with a speed equal to the

wave speed.

Standing Waves

• A standing wave does not

transfer Energy from one end

to the other.

• The two waves that form

it would individually carry

equal amounts of power

in opposite direction.

• A standing wave can only

exist if its wavelength satisfies

Fundamental Frequencies, Harmonics and

Overtones

o This is the fundamental frequency. The smallest frequency that corresponds

to the largest wavelength 2L.

o The other frequencies, f2, f3, f4, …. Are all multiple integer multiples of the

fundamental frequency. These frequencies are called harmonics. (shown

below)

o f2, f3, f4 and so on are what musicians call overtones. The second harmonic

f2 is called the first overtone.