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Figure shows a graph with sine theta on the y axis and theta on the x axis. It appears like a transverse wave with its y value varying from -1 to +1. The wave has crests at values theta equal to pi by 2, 5 pi by 2 and so on. It crosses the x axis at 0, pi, 2 pi and so on.
A sine function oscillates between + 1 and −1 every 2 π radians.

To construct our model of the wave using a periodic function, consider the ratio of the angle and the position,

θ x = 2 π λ , θ = 2 π λ x .

Using θ = 2 π λ x and multiplying the sine function by the amplitude A , we can now model the y -position of the string as a function of the position x :

y ( x ) = A sin ( 2 π λ x ) .

The wave on the string travels in the positive x -direction with a constant velocity v , and moves a distance vt in a time t . The wave function can now be defined by

y ( x , t ) = A sin ( 2 π λ ( x v t ) ) .

It is often convenient to rewrite this wave function in a more compact form. Multiplying through by the ratio 2 π λ leads to the equation

y ( x , t ) = A sin ( 2 π λ x 2 π λ v t ) .

The value 2 π λ is defined as the wave number    . The symbol for the wave number is k and has units of inverse meters, m −1 :

k 2 π λ

Recall from Oscillations that the angular frequency    is defined as ω 2 π T . The second term of the wave function becomes

2 π λ v t = 2 π λ ( λ T ) t = 2 π T t = ω t .

The wave function for a simple harmonic wave on a string reduces to

y ( x , t ) = A sin ( k x ω t ) ,

where A is the amplitude, k = 2 π λ is the wave number, ω = 2 π T is the angular frequency, the minus sign is for waves moving in the positive x -direction, and the plus sign is for waves moving in the negative x -direction. The velocity of the wave is equal to

v = λ T = λ T ( 2 π 2 π ) = ω k .

Think back to our discussion of a mass on a spring, when the position of the mass was modeled as x ( t ) = A cos ( ω t + ϕ ) . The angle ϕ is a phase shift, added to allow for the fact that the mass may have initial conditions other than x = + A and v = 0 . For similar reasons, the initial phase is added to the wave function. The wave function modeling a sinusoidal wave, allowing for an initial phase shift ϕ , is

y ( x , t ) = A sin ( k x ω t + ϕ )

The value

( k x ω t + ϕ )

is known as the phase of the wave , where ϕ is the initial phase of the wave function. Whether the temporal term ω t is negative or positive depends on the direction of the wave. First consider the minus sign for a wave with an initial phase equal to zero ( ϕ = 0 ) . The phase of the wave would be ( k x ω t ) . Consider following a point on a wave, such as a crest. A crest will occur when sin ( k x ω t ) = 1.00 , that is, when k x ω t = n π + π 2 , for any integral value of n . For instance, one particular crest occurs at k x ω t = π 2 . As the wave moves, time increases and x must also increase to keep the phase equal to π 2 . Therefore, the minus sign is for a wave moving in the positive x -direction. Using the plus sign, k x + ω t = π 2 . As time increases, x must decrease to keep the phase equal to π 2 . The plus sign is used for waves moving in the negative x -direction. In summary, y ( x , t ) = A sin ( k x ω t + ϕ ) models a wave moving in the positive x -direction and y ( x , t ) = A sin ( k x + ω t + ϕ ) models a wave moving in the negative x -direction.

[link] is known as a simple harmonic wave function. A wave function is any function such that f ( x , t ) = f ( x v t ) . Later in this chapter, we will see that it is a solution to the linear wave equation. Note that y ( x , t ) = A cos ( k x + ω t + ϕ ) works equally well because it corresponds to a different phase shift ϕ = ϕ π 2 .

Problem-solving strategy: finding the characteristics of a sinusoidal wave

  1. To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form y ( x , t ) = A sin ( k x ω t + ϕ ) .
  2. The amplitude can be read straight from the equation and is equal to A .
  3. The period of the wave can be derived from the angular frequency ( T = 2 π ω ) .
  4. The frequency can be found using f = 1 T .
  5. The wavelength can be found using the wave number ( λ = 2 π k ) .

Questions & Answers

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
An inherent property by virtue of which the body remains in its pure state or initial state
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
the flow of current is not current
bcoz it doesn't satisfy the algabric laws of vectors
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
hello Lydia.
What is momentum
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
is the eye the same like the camera
I can't understand
same here please
I think the question is that ,,, the working principal of eye and camera same or not?
yes i think is same as the camera
what are the dimensions of surface tension
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
what is the formula for pressure?
Goodness Reply
force is newtom
and area is meter squared
so in SI units pressure is N/m^2
In customary United States units pressure is lb/in^2. pound per square inch
Practice Key Terms 4

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