Definition of waves
A wave is a disturbance which travels through a medium and transfers energy from one point to another without causing any permanent displacement of the medium itself.
Types of waves
There are two types of waves, which are:
(A) transverse and longitudinal waves
Transverse wave: a wave is transverse if the direction of travel of the wave is perpendicular to the direction of vibration of the medium. For example, water waves and waves generated by plucking a string.
Longitudinal wave: longitudinal wave is a wave that travels in the same direction as the direction of vibration of the medium. For example, sound waves are longitudinal waves.
Mode of propagation of transverse waves
A wave can be generated by dropping a stone in water, and a small piece of light wood or cork is floating in the path of the wave. The cork will be seen to move up and down, that is, vibrate, or oscillate, in one spot as the wave travels past. This means that all the particles of water vibrate up and down as the wave passes. This up and down motion is perpendicular to the direction of the wave along the surface.
The upper curve is the crest while the lower part is the trough
Mode of propagation of longitudinal wave
Here, the particles vibrates behaving like a spiral spring that has a series of compressed regions and spaced out regions travelling along it. These are referred to as compressions and rarefactions.
(B) equation of a travelling wave
The equation of travelling wave can be written like a some or cosine functions:
O. Or y=cos O
Sine and cosine functions
In general, a wave whose amplitude is A and constant angular Velocity is w can be written as. y= A sin (wt-¢)
The constant angular distance ¢ which is known as phase constant can be related to the linear distance x by. ¢= 2πx/∆
Where ∆ is the wavelength of the wave, the quantity 2π/∆. is called wave number k.
Then we can conclude on the equation
y= A sin 2π. (vt-x)/ ∆
1.which of the following waves are longitudinal waves? (Wassce 1992)
l .Ripples on the surface of water
II. Waves produced by tuning fork vibrating in air
III. Light waves IV. Waves produced by a flute.
A. I and II only
B. I and III only
C. II and III only
D. II and IV only
E. III and IV only
Solution: longitudinal waves have to do with wave travelling in the same direction as the direction of a vibrating wave which makes II and IV a longitudinal wave.
2.the diagram below represents the profile of transverse waves. Which of the following points are in phase? (Wassce 1997)A. O and P
B. O and Q
C. O and R
D. O and S
E. O and T
3. A wave has an amplitude equal to 4.0m, angular speed ⅓ π rad 5-1 and phase angle ⅔ π rad. The displacement y of the wave particle is given as. (Wassce 2000)
A. y= 4sinπ/3 (t+2)
B. y= 4sinπ/3 (t+2/π)
C. y= 4sinπ/3 (2t+1)
D. y= 4sinπ/3 (t+2)
4. What type of wave is emitted by a loudspeaker? (Wassce 2015)
Solution: loudspeaker has to do with the emission of sound which is an example of a longitudinal wave
5. The equation of a wave is y= 0.005sin[x(0.5x-200t)] where x and y are in meters and t is in seconds. What is the Velocity of the wave? (Jamb 1992)
Solution: compare y=A sin(kx-wt) with y=0.005sin(0.5x-200t)
Now, 200 = 2πf, f=100/π
But, ∆= 2π/0.5
v= 110/∆ (2π)/(0.5) =400m/s
6. The equation of transverse wave travelling along a string is given by y= 0.3 sin(0.5x-50t) where y and x are in cm and t is in seconds. Find the maximum displacement of the particles from the equilibrium position. (Jamb 1994)
Solution: y= 0.3sin(0.5x-50t) compare with, y= Asin(kx-wt); A= 0.3m
7. If a sound wave goes from a cold-air region to a hot-air region, it’s wavelength be? (Jamb 1999)
C. Decrease then increase
D. Remain constant
8. The equation of a wave travelling along the positive x-direction is given by y = 0.25×10-3 sin (500t-0.025x). Determine the angular frequency of the wave motion. (Jamb 1999)
A. 0.25×10-3 rad/s
B. 0.25×10-1 rad/s
C. 5.00×102 rad/s
D. 2.50×103 rad/s
Solution: y= 0.25×10-3sin(500t-0.025x);
Compare with, y= Asin (wt+/-kx); w=500rad/s