1. 1 Waves 4 Lecture 4 - Stretched string Transverse waves on a stretched string D Aims: ëDerivation of the wave equation: > obtain the wave velocity, etc. ëWave impedance: > characteristic impedance of a stretched string. ëPolarization. > Linear; > circular; > Polarization and coherence.

2. 2 Waves 4 D General approach: ëConsider a small segment; ëFind the difference in forces on the two ends; ëFind the element’s response to this imbalance. D We need the following useful result: ë easy to see graphically: D Waves on a string: ëNeglect extension of the string i.e. tension is constant everywhere and unaltered by the wave. ëDisplacements are small so   <<.   x and  are << 1. ëNeglect gravity Derivation of the wave equation

3. 3 Waves 4 Stretched string D Forces on an element of the string: ëNet force in the y-direction is But so ëNet transverse force = ëTo apply Newton’s 2nd law we need mass and acceleration. These are: mass of element = acceleration (transverse) =

4. 4 Waves 4 Derivation of wave equation D Newton’s second law: ëThe wave velocity is not the transverse velocity of the string, which is D Two Polarization's D Two Polarization's for a transverse wave: > Horizontal (z) ; Vertical (y). mass acceleration force

5. 5 Waves 4 Wave impedance D The general concept: ëApplying a force to a wave medium results in a response and we can therefore define an impedance = applied force/velocity response. ëExpect 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. D Characteristic impedance: ëTransverse driving force: ëTransverse velocity:

6. 6 Waves 4 Impedance of a stretched string D Impedance ëFor a wave in +ve x-direction, recall ëImpedance (for wave in +ve x-direction)  Wave in -ve x-direction, Z has the opposite sign: ëNote that the impedance is real. i.e. the medium is lossless (in this idealised picture).

7. 7 Waves 4 Energy in a travelling wave D Energy is both kinetic and potential. ëKinetic energy associated with the velocity of elements on the string. ëPotential energy associated with the elastic energy of stretching during the motion. D Potential energy density D Potential energy density (P.E./unit length)  Calculate the work done increasing the length of segment  x against a constant tension T. ëIncrease in length of segment is

8. 8 Waves 4 Energy density D Increase in P.E D Increase in P.E (= force x extension) D Potential energy density ë D Kinetic energy density  K.E. of length  x ëDensity ëNote: instantaneous KE and PE are equal since D Total energy density ëSum of KE and PE extension force 1/2 m m v2v2 v2v2

9. 9 Waves 4 Energy: harmonic wave D Average energy density D Average energy density (section 1.1.3) ë ëaverage energy density is  consider each element, length  x, as an oscillator with energy  In time  t we excite a length  x=v  t. So the energy input is  Since Z=  v, D Mean power