Refraction of Mechanical
Waves
Ck12 Science
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Printed: March 26, 2014
AUTHOR
Ck12 Science
www.ck12.org
C HAPTER
Chapter 1. Refraction of Mechanical Waves
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Refraction of Mechanical
Waves
• Define and describe refraction of mechanical waves.
• State and use the law of refraction.
A straw in a glass of water seen from the side often appears broken, even though it is not. The apparent break is due
to the bending of light rays leaving the straw; as the light passes from the water to the glass and from the glass to the
air, the light rays are bent. Nonetheless, your eye traces the light ray backward as if the light has followed a straight
path from its origin at the straw. Since the light appears to have come from a different place, your eye sees the straw
as being broken.
Refraction of Mechanical Waves
When any wave strikes a boundary between media, some of the energy is reflected and some is transmitted. When
the wave strikes the media interface at an angle, the transmitted wave will move in a slightly different direction than
the incident wave. This phenomenon is known as refraction.
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Consider the image sketched above. Suppose that the waves represented here are water waves. The wave crests
are represented by the black lines in the image. As such, the distance between two consecutive black lines is the
wavelength. Let the red line represent a transition from deep to shallow water. This transition is called the media
interface. As the waves hit the boundary, the waves slow down. The right side of the wave reaches the boundary
before the left side of the wave, causing the left side to catch up and the angle of propagation to change slightly. This
change in direction can be seen in the yellow line, which is slightly angled at the boundary.
The refraction of waves across boundaries operates similarly to the method by which tanks are steered. Tanks do
not have a steering wheel. Instead, they have an accelerator to produce forward motion and separate brakes on each
tread. The operator uses brakes on both treads at the same time in order to stop, but brakes on only one tread to turn
the tank. By braking one side, the operator causes that side to slow down or stop while the other side continues at
the previous speed, causing the tank to turn towards the slower tread.
This sketch shows a wave ray striking an interface between old medium and new medium. A normal line has been
drawn as a dotted line perpendicular to the interface. The angle between the incident ray and the normal line is called
the angle of incidence, shown as θi , and the angle between the refracted ray and the normal line is called the angle
of refraction, θr .
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Chapter 1. Refraction of Mechanical Waves
We already understand that the change in the wave direction at the border depends on the difference between the two
velocities. This relationship is conveniently expressed in a mathematical relationship:
sin θr
vr
λr
= =
sin θi
vi
λi
The ratio of the sine of the angle of refraction to the sine of the angle of incidence is the same as the ratio of the
velocity of the wave in the new medium to the velocity of the wave in the old medium and equal to the ratio of
wavelength (λ) in the old medium to the wavelength in the new medium.
Example Problem: A water wave with a wavelength of 3.00 m is traveling in deep water at 16.0 m/s. The wave
strikes a sharp interface with shallow water with an angle of incidence of 53.0◦ . The wave refracts into the shallow
water with an angle of refraction of 30.0◦ . What is the velocity of the wave in shallow water and what is its
wavelength in the new medium?
Solution:
sin θr
sin θi
vr
vi
10.0 m/s
16.0 m/s
=
λr
λi
so
=
vr
vi
so
=
sin 30◦
sin 53◦
λr
3.00 m
=
vr
16.0 m/s
and vr = 10.0 m/s.
and λr = 1.88 m.
Example Problem: The ratio of the sin θr to sin θi is 0.769 . If the wavelength of a wave in a new medium is
5.00 × 10−9 m, what is its wavelength in the original medium?
Solution:
0.769 =
5.00 × 10−9 m
λr
so λi =
= 6.50 × 10−9 m
λi
0.769
Summary
• When any wave strikes a boundary between media, some of the energy is reflected and some is transmitted.
• When a wave strikes the media interface at an angle, the transmitted wave will move in a different direction
than the incident wave. This phenomenon is known as refraction.
θr
vr
λr
• At any media interface, sin
sin θi = vi = λi
Practice
http://www.youtube.com/watch?v=mH9VwivqjmE
Follow up questions.
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1. What causes refraction?
2. What doesn’t change during refraction?
Review
1. A laser beam passes through water and enters a glass block at an angle. The ratio of the speed of the wave
in glass to the speed in water is 0.866 . If the angle of incidence to the interface is 60◦ , what is the angle of
refraction?
2. A ray of light is traveling from air into glass at an angle of 30.0◦ to the normal line. The speed of the light in
air is 3.00 × 108 m/s and in glass the speed drops to 2.00 × 108 m/s. What is the angle of refraction?
3. Which of the following change when a water wave moves across a boundary at an angle between deep water
and shallow water?
a.
b.
c.
d.
e.
frequency
wavelength
speed
wave direction
period
4. Which of the following change when a water wave moves across a boundary exactly along the media interface
between deep water and shallow water?
a.
b.
c.
d.
e.
frequency
wavelength
speed
wave direction
period
5. The speed of sound is 340 m/s. A particular sound wave has a frequency of 320. Hz.
a. What is the wavelength of this sound in air?
b. If this sound refracts into water where the speed of sound is 4 times faster, what will be the new
wavelength?
c. What will be the new frequency?
6. When a light ray passes from air into diamond, the angle of incidence is 45.0◦ and the angle of refraction is
16.7◦ . If the speed of light in air is 3.00 × 108 m/s, what is the speed of light in diamond?
• refraction: The turning or bending of a wave direction when it passes from one medium to another of different
density.
References
1. Image copyright cheyennezj, 2013. http://www.shutterstock.com . Used under license from Shutterstock.com
2. CK-12 Foundation - Samantha Bacic. . CC-BY-NC-SA 3.0
3. Greg Goebel. http://www.public-domain-image.com/full-image/transportation-vehicles-public-domain-imag
es-pictures/tanks-public-domain-images-pictures/m7-priest-self-propelled-105-millimeter-howitzer-tank.jpg-co
pyright-friendly-image.html . Public Domain
4. CK-12 Foundation - Samantha Bacic. . CC-BY-NC-SA 3.0
4