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Velocity and acceleration of the medium

As seen in [link] , the wave speed is constant and represents the speed of the wave as it propagates through the medium, not the speed of the particles that make up the medium. The particles of the medium oscillate around an equilibrium position as the wave propagates through the medium. In the case of the transverse wave propagating in the x -direction, the particles oscillate up and down in the y -direction, perpendicular to the motion of the wave. The velocity of the particles of the medium is not constant, which means there is an acceleration. The velocity of the medium, which is perpendicular to the wave velocity in a transverse wave, can be found by taking the partial derivative of the position equation with respect to time. The partial derivative is found by taking the derivative of the function, treating all variables as constants, except for the variable in question. In the case of the partial derivative with respect to time t , the position x is treated as a constant. Although this may sound strange if you haven’t seen it before, the object of this exercise is to find the transverse velocity at a point, so in this sense, the x -position is not changing. We have

y ( x , t ) = A sin ( k x ω t + ϕ ) v y ( x , t ) = y ( x , t ) t = t ( A sin ( k x ω t + ϕ ) ) = A ω cos ( k x ω t + ϕ ) = v y max cos ( k x ω t + ϕ ) .

The magnitude of the maximum velocity of the medium is | v y max | = A ω . This may look familiar from the Oscillations and a mass on a spring.

We can find the acceleration of the medium by taking the partial derivative of the velocity equation with respect to time,

a y ( x , t ) = v y t = t ( A ω cos ( k x ω t + ϕ ) ) = A ω 2 sin ( k x ω t + ϕ ) = a y max sin ( k x ω t + ϕ ) .

The magnitude of the maximum acceleration is | a y max | = A ω 2 . The particles of the medium, or the mass elements, oscillate in simple harmonic motion for a mechanical wave.

The linear wave equation

We have just determined the velocity of the medium at a position x by taking the partial derivative, with respect to time, of the position y . For a transverse wave, this velocity is perpendicular to the direction of propagation of the wave. We found the acceleration by taking the partial derivative, with respect to time, of the velocity, which is the second time derivative of the position:

a y ( x , t ) = 2 y ( x . t ) t 2 = 2 t 2 ( A sin ( k x ω t + ϕ ) ) = A ω 2 sin ( k x ω t + ϕ ) .

Now consider the partial derivatives with respect to the other variable, the position x , holding the time constant. The first derivative is the slope of the wave at a point x at a time t ,

slope = y ( x , t ) x = x ( A sin ( k x ω t + ϕ ) ) = A k cos ( k x ω t + ϕ ) .

The second partial derivative expresses how the slope of the wave changes with respect to position—in other words, the curvature of the wave, where

curvature = 2 y ( x , t ) x 2 = 2 2 x ( A sin ( k x ω t + ϕ ) ) = A k 2 sin ( k x ω t + ϕ ) .

The ratio of the acceleration and the curvature leads to a very important relationship in physics known as the linear wave equation    . Taking the ratio and using the equation v = ω / k yields the linear wave equation (also known simply as the wave equation or the equation of a vibrating string),

2 y ( x , t ) t 2 2 y ( x , t ) x 2 = A ω 2 sin ( k x ω t + ϕ ) A k 2 sin ( k x ω t + ϕ ) = ω 2 k 2 = v 2 ,

Questions & Answers

state the properties of a base
Muhammad Reply
state 3 properties of a base
Muhammad
State three properties of a base
mubarak
a stone is projected from the top of the tower 24.5m high with velocity of 39.2m/s at an angle of 30° with horizontal compute time taken to hit he ground
Sarah Reply
it took a car driver 30mins to travel a distance of 50km.what is the average speed
Isiguzo Reply
Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
Louise Reply
Why is popo less than atmospheric? Why is popo greater than pipi?
Louise
The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.
Louise
David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.
Louise Reply
the pressure differential exerted a force on the bag greater than the gravitational force holding it on the floor.
gtitboi
what is angular velocity
Sthandazile Reply
The rate of change in angular displacement is defined as angular velocity.
Manorama
a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
Nicole Reply
What is the answer please
Mustapha
it took a car driver 30 mins to travel a distance of 50km what is the average speed .
Isiguzo
If centripetal force is directed towards the center,why do you feel that you're thrown away from the center as a car goes around a curve? Explain
Maira Reply
if there is a centripetal force it means that there's also a centripetal acceleration, getting back to your question, just imagine what happens if you pull out of a car when it's quickly moving or when you try to stop when you are running fast, anyway, we notice that there's always a certain force..
Lindomar
... that tends to fight for its previous direction when you try to attribute to it an opposite one ou try to stop it.The same thing also happens whe a car goes around a curve, the car it self is designed to a"straight line"(look at the position of its tyres, mainly the back side ones), so...
Lindomar
... whenever it goes around a curve, it tends to throw away its the occupiers, it's given to the fact that it must interrupt its initial direction and take a new one.
Lindomar
Which kind of wave does wind form
Matthias Reply
calculate the distance you will travel if you mantain an average speed of 10N m/s for 40 second
Abdulai Reply
400m/s
Feng
hw to calculate the momentum of the 2000.0 elephant change hunter at a speed of 7.50 m/s
Kingsley Reply
how many cm makes 1 inches
Hassan Reply
2.5
omwoyo
2.54cm=1inche
omwoyo
how do we convert from m/s to km/hr
Toni Reply
When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why?
SANA Reply
Explain why polarization does not occur in sound
Nuradeen
one ship sailing east with a speed of 7.5m/s passes a certain point at 8am and a second ship sailing north at the same speed passed the same point at 9.30am at what distance are they closet together and what is the distance between them then
Kuber Reply
Practice Key Terms 4

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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