<< Chapter < Page Chapter >> Page >

Solution

  1. Write the wave function of the second wave: y 2 ( x , t ) = A sin ( 2 k x + 2 ω t ) .
  2. Write the resulting wave function:
    y R ( x , t ) = y 1 ( x , t ) + y ( x , t ) = A sin ( k x ω t ) + A sin ( 2 k x + 2 ω t ) .
  3. Find the partial derivatives:
    y R ( x , t ) x = A k cos ( k x ω t ) + 2 A k cos ( 2 k x + 2 ω t ) , 2 y R ( x , t ) 2 x = A k 2 sin ( k x ω t ) 4 A k 2 sin ( 2 k x + 2 ω t ) , y R ( x , t ) t = A ω cos ( k x ω t ) + 2 A ω cos ( 2 k x + 2 ω t ) , 2 y R ( x , t ) 2 t = A ω 2 sin ( k x ω t ) 4 A ω 2 sin ( 2 k x + 2 ω t ) .
  4. Use the wave equation to find the velocity of the resulting wave:

    2 y ( x , t ) x 2 = 1 v 2 2 y ( x , t ) t 2 , A k 2 sin ( k x ω t ) 4 A k 2 sin ( 2 k x + 2 ω t ) = 1 v 2 ( A ω 2 sin ( k x ω t ) 4 A ω 2 sin ( 2 k x + 2 ω t ) ) , k 2 ( A sin ( k x ω t ) 4 A sin ( 2 k x + 2 ω t ) ) = ω 2 v 2 ( A sin ( k x ω t ) 4 A sin ( 2 k x + 2 ω t ) ) , k 2 = ω 2 v 2 , | v | = ω k .

Significance

The speed of the resulting wave is equal to the speed of the original waves ( v = ω k ) . We will show in the next section that the speed of a simple harmonic wave on a string depends on the tension in the string and the mass per length of the string. For this reason, it is not surprising that the component waves as well as the resultant wave all travel at the same speed.

Check Your Understanding The wave equation 2 y ( x , t ) x 2 = 1 v 2 2 y ( x , t ) t 2 works for any wave of the form y ( x , t ) = f ( x v t ) . In the previous section, we stated that a cosine function could also be used to model a simple harmonic mechanical wave. Check if the wave

y ( x , t ) = 0.50 m cos ( 0.20 π m −1 x 4.00 π s −1 t + π 10 )

is a solution to the wave equation.

This wave, with amplitude A = 0.5 m , wavelength λ = 10.00 m , period T = 0.50 s , is a solution to the wave equation with a wave velocity v = 20.00 m/s .

Got questions? Get instant answers now!

Any disturbance that complies with the wave equation can propagate as a wave moving along the x -axis with a wave speed v . It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.

Summary

  • A wave is an oscillation (of a physical quantity) that travels through a medium, accompanied by a transfer of energy. Energy transfers from one point to another in the direction of the wave motion. The particles of the medium oscillate up and down, back and forth, or both up and down and back and forth, around an equilibrium position.
  • A snapshot of a sinusoidal wave at time t = 0.00 s can be modeled as a function of position. Two examples of such functions are y ( x ) = A sin ( k x + ϕ ) and y ( x ) = A cos ( k x + ϕ ) .
  • Given a function of a wave that is a snapshot of the wave, and is only a function of the position x , the motion of the pulse or wave moving at a constant velocity can be modeled with the function, replacing x with x v t . The minus sign is for motion in the positive direction and the plus sign for the negative direction.
  • The wave function is given by y ( x , t ) = A sin ( k x ω t + ϕ ) where k = 2 π / λ is defined as the wave number, ω = 2 π / T is the angular frequency, and ϕ is the phase shift.
  • The wave moves with a constant velocity v w , where the particles of the medium oscillate about an equilibrium position. The constant velocity of a wave can be found by v = λ T = ω k .

Conceptual questions

If you were to shake the end of a taut spring up and down 10 times a second, what would be the frequency and the period of the sinusoidal wave produced on the spring?

Got questions? Get instant answers now!
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