Intro

Question 1

transverse waves travelling in the x-directions can be described mathematically as

Answer 1

y(x,t)=Acos(kx +/- wt)

Question 2

what is a wave

Answer 2

it is a collective bulk disturbance that propagates through a medium, in which whatever happens at any specific point is a delayed
response to the disturbance at adjacent points.

Question 3

what do waves carry

Answer 3

The components of the medium have no net displacement because of the wave passing – it is only energy that a wave carries.

Question 4

assumptions made for transverse waves on a string

Answer 4

1. the tension is high enough that any forces acting on elements of the string can be considered as a consequence of only the tension (ignore gravity)
2. the tension does not vary significantly when the string is undergoing small transverse displacements form rest

Question 5

If we assume that the shape of the pulse remains constant, then the speed of propagation is determined by

Answer 5

the tension and the string’s linear density

Question 6

relationship between c, T and p

Answer 6

from dimensional analysis

c = sqrt(T/p)

Question 7

to consider the microscopic picture, we consider

Answer 7

a small element of the stretched string

Question 8

microscopic picture - we know that

Answer 8

1. any element is only moving in a transverse direction, not along the string itself.
2. only reason it moves is unbalanced force which must be in transverse direction

Question 9

microscopic picture - unbalanced force in the transverse direction

Answer 9

already discounted gravity

the force can only come from the tension in the string

Question 10

macroscopic - assumptions

Answer 10

pulse shape is constant

tension or linear mass density determines propagation speed

Question 11

linear mass density

Answer 11

mass per unit length

Question 12

wave speed increases when

Answer 12

tension increases (strong forces)

linear mass density decreases (easier to move)

Question 13

deriving the wave equation - set up

Answer 13

element of string of length x

string at rest
x0=centre

Question 14

deriving the wave equation - first

Answer 14

decompose tension into vertical and horizontal components

Question 15

wave equation - small displacements so

Answer 15

small angle

use small angle approx

Question 16

deriving wave equation - longitudinal

Answer 16

net force: Tcos(θ2)-Tcos(θ1)

cosθ approx =1

so F=T-T=0

no net horizontal force

Question 17

deriving wave equation - transverse step 1

Answer 17

gradients at both ends

dy/dx at x0-dx/2 = tan(θ1)

dy/dx at x0+dx/2 = tan(θ2)

Question 18

deriving wave equation - net transverse force

Answer 18

F=Tsin(θ2)-Tsin(θ1)

approx = Ttan(θ2)-Ttan(θ1)

sub in gradients

so F=T(grad1 - grad2)

Question 19

elements will feel net transverse force if

Answer 19

gradients at the two ends are different

Question 20

the net transverse force acts on the mass element and produces

Answer 20

an acceleration
d2y/dt2 at x0

Question 21

let the string have uniform mass density so the mass of a string element is

Answer 21

pdx

Question 22

how to get to wave equation

Answer 22

from F=ma

F form transverse force
m from pdx
a from d2y/dt2 @ x0

Question 23

ignored so far in the wave equation

Answer 23

frictional losses both internal and due to surrounding medium

Question 24

the derived equation is linear so

Answer 24

the superposition principle holds

Question 25

superposition principle

Answer 25

if we have solutions y1(x,t) and y2(x,t) then any combination Ay1(x,t) +/- By2(x,t) will also be a solution

Question 26

the three distinct velocities in wave motion

Answer 26

1. particle velocity 2. the wave or phase velocity 3. the group velocity

Question 27

particle velocity

Answer 27

the simple harmonic velocity of the oscillator about its equilibrium position

Question 28

wave or phase velocity

Answer 28

velocity with which planes of equal phase, crest or troughs, progress through the medium

Question 29

group velocity

Answer 29

when a number of waves of different frequencies, wavelengths and velocities are superimposed to form a group

Question 30

reference frames - let the x' frame travel to

Answer 30

the right, with the time origin being when two frames coincide

Question 31

reference frames - let xf be

Answer 31

the location of the origin of the moving frame (x') in the stationary frame (x)

Question 32

reference frames - we can say that xf=

Answer 32

ct

Question 33

the coordinates of xf are either

Answer 33

(x,y) or (x'y') depending on the frame by y=y'

Question 34

y=y' so x=

Answer 34

x'+xf =x'+ct x'=x-ct

Question 35

for a pulse moving to the right with y=y(x') - time independent

Answer 35

y(x,t) = y(x') = y(x-ct)

Question 36

for a pulse moving to the left (-ve x direction)

Answer 36

y(x,t)=y(x+ct)

Question 37

if the pulse moves in the +ve x direction, the transverse displacement of the string can be expressed as

Answer 37

a function of the single composite variable x-ct

Question 38

if the pulse moves in the -ve x-direction, the transverse displacement of the string can be expressed as

Answer 38

a function of the single composite variable x+ct

Question 39

how to check if something is a solution to the wave equation

Answer 39

find the first and second spatial and temporal derivatives and compare may need to use the chain rule