<< Chapter < Page Chapter >> Page >
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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask