Transverse Waves

Question 1

Wave number equation and units

Answer 1

k=2pi/ lambda [m^-1]

Question 2

Angular frequency units

Answer 2

rads^-1

Question 3

Type of transverse wave and longitudinal

Answer 3

Longitudinal= sound
Transverse= string or electromagnetic

Question 4

Wave speed equation generally

Answer 4

C=f lambda

Question 5

Phase velocity equation

Answer 5

c= omega/k

Question 6

Wave equation

Answer 6

c^2 partial second dy/dx = partial second dy/dt

Question 7

D’Alembert solution
Direction

Answer 7

y(x,t) = f(x plus or minus ct)
Minus to the right differential of phi by dt =0 solve for v_p is positive where phi =kx - omega t

Question 8

String c

Answer 8

c= root T/ mu

Question 9

Special case for sine wave velocities

Answer 9

V_p=V_g
Phase velocity= group velocity
Every point exhibits SHO

Question 10

Superposition principle

Answer 10

If 2 or more waves meet in a region of space, then at each instant of time the net disturbance they cause at any point is equal to the sum of the disturbances of each wave.

Question 11

Superposition principle using L operators

Answer 11

L= wave equation, Ly_1 =0 and Ly_2 =0 because solutions
L(a_1y_1 +a_2y_2) = a_1L(y_1) + a_2L(y_2)=0 so solution

Question 12

Derivation of c=root T/mu

Answer 12

a«lambda
Use small approximation makes triangle delta L = delta x Horizontal F=0
Vertical F= -Tsin theta + Tsin(theta + delta theta)= -Ttan theta+ T tan(theta+delta theta)
=T second dy/dx deltax Use F=ma and delta m= mu delta x compare to wave equation

Question 13

Superposition examples

Answer 13

Diffraction patterns
Magnetic fields

Question 14

Waves at a Boundary equations

Answer 14

y_i =ae^[i(k_1 x -wt’)]
y_r= re^[i(-k_1x-wt’)]
y_t=te^[i(k_2x-wt’)]

Question 15

Boundary conditions of 2 different string size

Answer 15

Vertical displacement continuity: at x=0 incident+reflected=transmitted a+r=t
Continuity of gradient, force: T i derivative + T r derivative = T t derivative all by x at x=0 k_1(a-r)=k_2t

Question 16

Combining boundary conditions r and t in terms of a

Answer 16

r=k1 -k2 / k1 +k2 a
t= 2k1/k1+k2 a

Question 17

Squiggly R and squiggly T

Answer 17

R= r/a = k1-k2/k1+k2 reflection coefficient
T=t/a=2k1/k1+k2 transmission coefficient

Question 18

Squiggly R and T relationship

Answer 18

1+R=T

Question 19

k_i to write squiggly R and T in terms of mu

Answer 19

=constant x root mu _i

Question 20

4 scenarios

Answer 20

Same material mus are equal No R T=1 just transmitted wave of same amplitude
mu2>mu1 R=negative pi phase change reflected wave T=positive so exists
mu2 tends to infinity (clamp) R tends to -1 and T=0 perfect inverted reflection no transmission
mu2 tends to 0 (free end) R=1 and T=2 perfect reflection no transmission double amplitude at free end.

Question 21

Impedance general equation in words

Answer 21

Z= driving force/ resulting speed

Question 22

For a string Z

Answer 22

Z=T/c = root(T mu)

Question 23

Squiggly R and T in terms of impedance

Answer 23

R= Z1-Z2/Z1+Z2
T=2Zq/Z1+Z2

Question 24

Impedance matching

Answer 24

Z1=Z2 improves transmission reduces reflection

Question 25

Electromagnet impedance

Answer 25

Z_i = root( mu/ epsilon) proportional to 1/n_i

Question 26

Kinetic energy in string

Answer 26

KE= 1/2 (mu delta x) (dy/dt) squared

Question 27

Potential energy in string

Answer 27

= work done =1/2 T delta x (dy/dx)squared

Question 28

Total energy in a string and at destructive superposition, normal pulse

Answer 28

KE= max No PE because no stretch Total = KE+PE KE=PE

Question 29

Total energy density

Answer 29

E= mu (dy/dt) squared

Question 30

Power for string and any wave

Answer 30

P= 1/2 mu omega squared A squared c from integral over 1 wavelength x 1/ lambda and energy/ unit time P= Z/2 omega squared A squared

Question 31

R and T energy reflection and transmission coefficients

Answer 31

R= P_r/P_i = squiggly R squared T= P_t/P_i = squiggly T squared x Z2/Z1

Question 32

R relation to T

Answer 32

R+T=1 due to energy conservation so Power cause = Power effect P_i=P_r+P_t

Question 33

Standing wave cause

Answer 33

2 equal amplitude travelling waves with the same f and Magda but moving in opposite directions superpose

Question 34

Standing wave amplitude

Answer 34

2a where a is amplitude of causing waves

Question 35

Superposition equation to create standing wave

Answer 35

y1= Ae^[i(kx-wt)] y2=Ae^[i(-kx-wt)] Add together = 2Acos(kx)[cos(wt)-isin(wt)] Separate time and x parts not travelling

Question 36

Distance between adjacent anti nodes or nodes

Answer 36

Lambda/ 2

Question 37

Scenario for standing wave

Answer 37

Fixed end= node Free end=anti-node

Question 38

Travelling vs standing wave amplitude, phase, wavelength and energy

Answer 38

T= constant amplitude, S= range 0 to 2a T=SHO phase is kx0, S phase either 0 or pi T=2pi/k, S=pi/k using adjacent nodes T= energy to direction of wave, S=no net transfer

Question 39

Quantised values: k, omega and f for harmonics

Answer 39

kn=nip/L Omegan=npic/L fn=no/2L L=n la,Bea/2 n is harmonic number

Question 40

Group velocity def, equation and found

Answer 40

Group of waves form wave packets/ pulse V_G = domega/dk Look at maxima motion can spread

Question 41

V_p V_G relations

Answer 41

Equal = non-dispersive Not equal= dispersive

Question 42

What doesV_p and V_G describe

Answer 42

V_G describes velocity of envelope V_P describes constituent wave speeds

Question 43

Wave packet equation

Answer 43

See notes end of transverse

Question 44

Fast oscillations and slow oscillations speeds

Answer 44

Fast=Vp Slow=Vg