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characteristics, properties
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What is a wave?
What is a wave?
A wave is a disturbance propagated from one
place to another with no actual transport of
matter
Classification of a wave
EM waves vs Mechanical waves
Mechanical wave
Motion requires a medium
Electromagnetic wave
Motion does not need a medium
Electromagnetic wave
Transverse wave
Displaces particles
perpendicular to the motion of the wave
Longitudinal wave
Cause the particles of a
medium to move parallel to the direction of
motion of the wave
Longitudinal wave
Has a crowded area causing a high-pressure region called
compression, and the opposite is
rarefaction
Periodic motion
The repetitive motion of a particle about the equilibrium position caused by a restoring
force when it is displaced
Simple Harmonic Motion (SHM)
• within the elastic limit, the distortion in an object is directly proportional to the distorting force
• If the restoring force obeys Hooke’s law, then the resulting vibration is called Simple Harmonic Motion (SHM)
Hooke’s law
Periodic Wave Characteristics
Wavelength (λ)The distance from crest to crest (or trough to trough); expressed
in meters
Periodic Wave Characteristics
Amplitude (A)
The distance of crest (or trough) from the midpoint of
the wave
Periodic Wave Characteristics
Frequency (f)The number of waves that
passed a fixed point per second; measured in hertz (Hz)
Periodic Wave Characteristics
Period (T)The time it takes a wave to travel a
distance equal to a wavelength; measured in second
T= 1/f
Periodic Wave Characteristics
Wave velocity (v)
Distance travelled by a wave crest in one period
v= λ/T
Periodic Wave Characteristics
Speed of a transverse wave (v)
F= tension μ= mass per unit length
v=√ (F/μ)
Periodic Wave Characteristics
Speed of a longitudinal wave (v)
B= bulk modulus of the mediumρ=density of the medium
v=√( B/ρ)
Periodic Wave Characteristics
Speed of a longitudinal wave (v)
Ƴ= Young’s modulus of the mediumρ=density of the medium
v=√ (Ƴ/ρ)
Example
The linear mass density of the clothesline is 0.250kg/m. How much tension does Rocky have to apply to produce the observed wave speed of 12.0m/s? F= 36.0N
Evaluate
Hold an alarm clock at arm’s length from your ear. While it’s still ringing , place the clock on the tabletop, press
your ear against the table. Which medium is more efficient in conducting
sound?
ExampleA hiker shouts on top of a mountain toward a vertical cliff, 688m away. The echo is heard 4s after. a) What is the speed of sound? b) The wavelength of the sound is 0.75m. What is its frequency? c) What is the period of the wave?
v=344.5m/s
f=458.67/s
T= 2.18x10-3s
remember
The displacement (y) of a wave is always a function of x and t
remember
k= wave numberk= 2π/λ
Note: Wave is moving to the +x
Power of a Wave
The rate of energy transfer (Pav)
ω= angular frequency; A= amplitude
ω=2πf
𝑃𝑎𝑣=12
√𝜇 𝐹𝜔2 𝐴2
Intensity of a Wave
The average power per unit area
For fluids in a pipe
For a solid rod
Example: Work as a group
a)In the previous example, if the amplitude of the wave is 0.075m and f=2Hz at what maximum rate does Rocky put energy into the rope? That is, what is his max. instantaneous power? b) What is his ave. power? c) As Rocky tires, the amplitude decreases. What is the average power when the amplitude has dropped to 7.50mm?
Pmax=2.83 W
Pav=1.415 W
Pav= 0.01415W
Periodic wave phenomena
Displacement y as a function of x and t
Wave reflects from a fixed end
Wave reflects from a free end
Boundary Conditions
the conditions at the end of the string, such as a rigid support or the complete
absence of transverse force
The total displacement at pt. O is zero at all times
The total displacement at pt. O is not zero but the slope is always zero.
The Principle of Superposition
When two waves overlap, the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if
only the first wave were present and the displacement it would have if only the
second wave were present.
The wave pattern doesn’t appear to be moving in either direction along the string
Nodes are points at which the string
never moves
Antinodes are points at which the amplitude of
string motion is greatest
Constructive interference
Destructive interference
Zero displacement
Large resultant displacement
The wave function for a standing wave
Sum of the individual wave function
or
Adjacent nodes are one half wavelength
apart
So the length of the string must be:
n= 1, 2, 3, …..For strings fixed at
both ends
Normal Modes on a String
N NAl
l =λ/2
λ =2l
Normal Modes on a String
N N
lAA
Nl =λ
Normal Modes on a String
N N
lA
N
l =3λ/2
NA A
λ =2l / 3
Normal Modes on a String
A A AAN N N N N
l
l =2λ
λ = l /2
Normal Modes on a String
For an oscillating system, it is a motion in which all particles of
the system move sinusoidally with the
same frequency
Fundamental frequency
The smallest frequency that corresponds to the largest
wavelength; n=1
SW frequencies
The possible standing wave frequencies
n= 1, 2, 3, …..For strings fixed at
both ends
Harmonic series
The series of possible standing wave frequencies (>f1)
Note: Musicians call these frequencies overtonesf2 is the second harmonic or the first overtone
f3 is the third harmonic or the second overtone
Standing waves and String Instruments
Shows the inverse dependency of frequency on length
Example
In an effort to get your name in the Guinness Book of World Records, you set out to build a bass viol with strings that have a length of 5.00m between fixed points. One string has a linear mass density of 49.0g/m and a fundamental frequency of 20.0Hz (the lowest frequency that the human ear can hear). Calculate (a) the tension of this string, (b) the frequency and the wavelength on the string of the second harmonic, and (c) the frequency and wavelength on the string of the second overtone.
Wave front
• It is the locus of all adjacent points on a wave that are in phase
• Phases are points on successive wave cycles of a periodic wave that are displaced from their rest position by the same amount in the same direction and are moving in the same direction
Interference
Superposition of two waves as they come in contact with
each other
Huygens Principle
States that waves spreading out from a point source may be regarded as the overlapping of tiny secondary wavelets and that every point on any wave front may be regarded as a new point source of secondary wavelets
Constructive interference Destructive interference
Two interacting waves are in-phase
Two interacting waves are out-of-phase
Diffraction
Bending of waves upon
interacting with an obstacle