1 Waves 4

Lecture 4 - Stretched stringLecture 4 - Stretched string

Transverse waves on a stretched stringTransverse waves on a stretched string

Aims:Aims: Derivation of the wave equation:

obtain the wave velocity, etc. Wave impedance:

characteristic impedance of a stretched string.

Polarization. Linear; circular; Polarization and coherence.

2 Waves 4

General approach:General approach: Consider a small segment; Find the difference in forces on the two ends; Find the element’s response to this imbalance.

We need the following useful result:We need the following useful result:

easy to see graphically:

Waves on a string:Waves on a string: Neglect extension of the string i.e. tension is

constant everywhere and unaltered by the wave. Displacements are small so

<<. x and are << 1.

Neglect gravity

Derivation of the wave equationDerivation of the wave equation

xdxdg

xgxxg

3 Waves 4

Stretched stringStretched string

Forces on an element of the string:Forces on an element of the string:

Net force in the y-direction is

But

so

Net transverse force =

To apply Newton’s 2nd law we need mass and acceleration. These are:

mass of element =

acceleration (transverse) =

tantansinsin TTTT

xxxx

tan

xtan

xx

T

2

2

x22 t

4 Waves 4

Derivation of wave equationDerivation of wave equation

Newton’s second law:Newton’s second law:

The wave velocity is not the transverse velocity of the string, which is

Two Polarization's Two Polarization's for a transverse wave: Horizontal (z) ; Vertical (y).

2

2

2

2

txx

xT

T

tTx velocity wave

2

2

2

2

massmass accelerationaccelerationforceforce

.t

5 Waves 4

Wave impedanceWave impedance

The general concept:The general concept: Applying a force to a wave medium results in a

response and we can therefore define an impedance = applied force/velocity response.

Expect the impedance to be real (force and velocity are in-phase) - since energy fed into the medium propagates (without loss) away from the source of the excitation. Imaginary impedance - no energy can be

transported (examples later: waveguide below cut-off).

Complex impedance - lossy medium.

Characteristic impedance:Characteristic impedance:

Transverse driving force:

Transverse velocity:

xTTTF

tansin

t

6 Waves 4

Impedance of a stretched stringImpedance of a stretched string

ImpedanceImpedance

For a wave in +ve x-direction, recall

Impedance (for wave in +ve x-direction)

Wave in -ve x-direction, Z has the opposite sign:

Note that the impedance is real. i.e. the medium is lossless (in this idealised picture).

txT

Z

velocity transverseforce driving

tvx

tf

vtu

fx

1

;

so,

vTvTZ

vTvTZ

7 Waves 4

Energy in a travelling waveEnergy in a travelling wave

Energy is both kinetic and potential.Energy is both kinetic and potential. Kinetic energy associated with the velocity of

elements on the string. Potential energy associated with the elastic

energy of stretching during the motion.

Potential energy density Potential energy density (P.E./unit length) Calculate the work done increasing the length

of segment x against a constant tension T.

Increase in length of segment is

xx

xx

x

xxx

x

22

2122

21

21

1

8 Waves 4

Energy densityEnergy density

Increase in P.E Increase in P.E (= force x extension)

Potential energy densityPotential energy density

Kinetic energy densityKinetic energy density K.E. of length x

Density

Note: instantaneous KE and PE are equal since

Total energy densityTotal energy density

Sum of KE and PE

xx

T

2

21.

extensionextensionforceforce

2

21.

x

T

2

21

t

x

1/21/2 mm v2v2

2

21

t

21 vTtvx and

2

t

9 Waves 4

Energy: harmonic waveEnergy: harmonic wave

Average energy densityAverage energy density (section 1.1.3)

average energy density is

consider each element, length x, as an oscillator with energy

In time t we excite a length x=vt. So the energy input is

Since Z=v,

Mean powerMean power

2; okxti

o AeA

tt21

221

oA

221

oAx

221

oAtv

2221

oAZ