chapter 16: waves 1 1. the equation...a. la rosa lecture notes ph‐213 general physics chapter 16:...
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A. La Rosa Lecture Notes
PH‐213 GENERAL PHYSICS ________________________________________________________________________
CHAPTER 16: WAVES‐1
1. The wave equation
Outline of Mechanical waves Longitudinal and transverse waves
Waves in a string, sound waves
The wave equation Description of waves using functions of two variables
Travelling waves
The wave equation 0y
v
1y2
2
22
2
tx
Waves in a string Reflection and transmission of waves at an interface
Example A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. a) How far apart are two points x1 and x2 that, at a given time to,
differ in phase by /3 rad? b) What is the phase difference between two displacements Y1 and
Y2 at a certain point (at a certain xo point) at times 1 .0 ms apart? (Notice, you are not being asked for the values of Y1 nor Y2; just their relative phase difference.)
Solution
v = f f = 500 Hz and v = 350 m/s implies, = 350/500 = 0.7 m Y = A Cos (k x - t +
a) Things happen at t = t0 Phase1 = k x1 - t0 +
Phase2 = k x2 - t0 +
Phase1 - Phase2 = [k x1 - t0 + ] - [ k x2 - t0 +
= [k( x1 - x2 /3 rad = [k( x1 - x2 implies
x1 - x2 = (/3 )/k = (/3 ) /2 =
= 0.7/6 = 11.7 cm
b) Things happen at x = x0
Phase1 = k x0 - t1 +
Phase2 = k x0 - t2 +
Phase1 - Phase2 = [ k x0 - t1 + ] - [k x0 - t2 +
= [ ( t2 - t1
= [ f ( t2 - t1
( t2 - t1 1 ms implies,
Phase1 - Phase2 = [ 500 Hz ( 10-3s
Cos = 1 – (1/2) 2 + … For small : Cos ~ 1
Sin = – (1/3!) 3 + … For small : Sin =
Tan = – (1/3!)2 3 + … For small : Tan =
For the case of the string:
Horizontal force Fx = T Cos - T Cos
Vertical force Fy = T Sin - T Sin
For the case of small and smallcos = 1, cos = 1
sin = , sin =
tan = , tan =
Horizontal force Fx = T - T = 0
Vertical force Fy = T Sin - T Sin But, conveniently for the steps to follow below, we
expressed the expression above in terms of the tangent function
= T tan - T tan
The tan can be interpreted as the slope of the y vs x curve. This is illustrated in the figure below.
Y
x
y(x,t)
Notice, at a given fixed time t:
y
x x
= tan
Geometrical interpretation
Or,
Twhere
tx
v0
y
v
1y2
2
22
2
which is the wave equation
It admits solutions of the form
y(x,t) = f (x-vt) + g(x+vt)
where f and g are arbitrary functions
x)
At x=0
The little guy A shakes his hands UP and DOWN with a (temporal) frequency f. So, at x=0 the motion is described by Y = Cos (0-t) = Cos t = 2f
At x=xo
The little guy A sees that a particular segment of the string located at x=xo (point “P”) goes UP and DOWN with frequency f. y = Cos (k xo -t)
P
f = v
