1. Chapter-16
Waves-I

2. Chapter-16 Waves-I Topics to be studied: Types of waves.
Amplitude, phase, frequency, period, propagation speed of a wave
Mechanical waves propagating along a stretched string.
Wave equation
Principle of superposition of waves
Wave interference
Standing waves, resonance

3. Ch 16-2,3 Waves: Mechanical Waves
Wave Motion: Disturbance of particles of a medium
3 types Of Wave
1-Mechanical Waves: waves requires a medium to propagate, e.g. water & string waves
2-Electromagnetic waves : waves travel without a medium e.g. radio waves, etc
3-Matter waves: waves associated with atoms and subatomic particles
Mechanical Waves: Two types
1-Transverse Waves- particle displacement  to wave velocity direction e.g. string waves
2-Longitudinal Wave: Particle displacement  to wave velocity direction
e.g. Sound wave

4. Ch 16-4 Wavelength and Frequency
Equation of a Wave:
traveling towards right :
y1(x,t)=ymsin(kx-t)
traveling towards left:
y2(x,t)=ymsin(kx+t);
T is period of oscillations and related with angular frequency  by : =2/T
 is wavelength and related with angular wave number  by: k=2/

5. Checkpoint-Ch-16-1
The figure is composed of three snapshots, each of a wave traveling along a particular string. The phases for the waves are given by (a) 2x-4t (b) 4x-8t © 8x-16t.
Which phase corresponds to which wave?
Phase  = kx-t, where
wave number k= 2/
and wave length  = 2 /k.
The for (a)  = 2 /k = 
(b)  = 2 /k = /2
(c)  = 2 /k = /4
Which phase corresponds to which wave?
(a)=2
(b) = 3
( c) = 1

6. Ch16-5 Speed of a Traveling Wave
Wave travels a distance of x in time t, then wave speed v=dx/dt :
Since particle A retains its displacement, which means kx- t= constant
Then v=+dx/dt= /k
v =/T =f
Positive v for kx- t
Negative v for kx+ t

7. Ch16-6 Wave Speed on a stretched string
Net Force on small length l is FR ; where
FR=2( sin)=  (2)
s=R’ ; 2= l /R;
m =l; where  is linear mass density
FR= m v2/R= (l/R)
v2=(l/m )= /
v= /

8. Ch16-6 Transverse velocity of particles on a stretched string
Transverse velocity of string particles u=dy/dt
where y(x,t)=ymsin(kx-t);
u= -ymcos(kx-t);
Transverse acceleration ay= du/dt
=-2ymsin(kx-t)
=-2y ;
Maximum Transverse velocity um=-ym
Maximum Transverse acceleration ay-m=-2ym

9. Checkpoint-Ch-16-2 v=/; umax=-ym
Here are the equations of three waves.
(1) y(x,t)= 2 sin (4x-2t)
(2) y(x,t)= sin (3x-4t)
(3) y(x,t)= 2 sin (3x-3t)
Rank the waves according to their
(a) wave speed
(b) maximum transverse speed, greatest first.
v=/; umax=-ym
(1) v=-2/4=-0.5 m/s
umax =4 m/s
2) v=-4/3=-1.33 m/s
3) v=-3/3=-1.0 m/s
umax =6 m/s

10. Ch16-6 Energy and Power of a Wave traveling along a String
Kinetic energy K due to transverse velocity u and elastic potential energy U due to stretching of the string element
At point a, element at rest (turning point), y=U=0 and K=0.
At point b, element fully stretched y, U and K are max

11. Ch16-6 Energy and Power of a Wave traveling along a String
Rate of Energy Transmission:
Kinetic energy dK of a string segment with mass dm
dK=dmu2/2=dx[-ymcos(kx-t)]2/2
dK=2ym2cos(kx-t)2 dx/2
dK/dt=v2ym2cos(kx-t)2/2
(dK/dt)Avg=[v2ym2 cos(kx-t)2/2]Avg
[cos(kx-t)2]Avg=1/2
Then (dK/dt)Avg= v2ym2 /4
(dU/dt)Avg= (dK/dt)Avg
Pavg= 2(dK/dt)Avg=v2ym2 /4

12. Ch16-9 Principle of Superposition for Waves
Overlapping of two traveling waves y1(x,t) and y2(x,t) results in wave y’(x,t), the algebraic addition of the two waves:
y’(x,t)= y1(x,t)+y2(x,t)

13. Ch16-10 Interference of Waves
Interference : Resultant displacement of medium particle due to addition of two waves :
Two traveling waves:
y1 (x,t) = ymsin(kx- t) and
y2 (x,t) = ymsin(kx-t+ )
interfere to give resultant wave
y’(x,t)=ymsin(kx- t)+ymsin(kx- t+ )
sin + sin = 2sin{(+)/2}cos{(-)/2}
Then
y’(x,t)=[2ymcos(/2)] sin(kx- t+ /2)
y’(x,t)=y’m sin (kx-wt+ /2)
y’m= [2ymcos(/2)]

14. Ch16-12 Standing Waves
Standing Waves: Due to interference of two identical traveling waves moving in opposite direction
Fully constructive interference for time difference t= 0, T/2, T
Fully destructive interference for time difference t= T/4, 3T/4

15. Ch16-12 Standing Waves y’(x,t)=ymsin(kx-t)+ymsin(kx+t)
Two traveling waves moving in opposite direction :
y1 (x,t) = ymsin(kx-t) and y2 (x,t) = ymsin(kx-t) interfere to give resultant standing wave
y’(x,t)=ymsin(kx-t)+ymsin(kx+t)
sin + sin = 2sin{(+)/2}cos{(-)/2}
Then y’(x,t)= [2ym sin(kx)] cos(t) =y’m cos(t)
where y’m= [2ym sin(kx)] and it depends upon x
Nodes y’m=0 for kx=0,,..=n (n=0,1,2,..); x= n/k =n/2
Antinodes y’m=2ym for kx=/2,3/2..=(n+1/2); (n=0,1,2,..); then x= (n+1/2)/k = (n+1/2) /2

16. Ch16-13 Standing Waves and Resonances
For certain frequencies, called resonance frequencies, traveling waves interferes and produce very large amplitude.
The string will resonate if fixed end of the string are nodes:
The first mode is 1/2=L ; 2nd mode is
2 (2/2) =L ; 3rd mode is 3 (3/2) =L
Then nth mode n (n/2) =L (n=1,2,3,….)
n =2L/n but fn=v/ n ;
Then fn=nv/2L =n(v/2L)=nf1
n is called harmonic ; n=1 is called first harmonic and is called fundamental mode of oscillation and f1 is called fundamental frequency
n=2 second harmonic etc A B