7.5 Worksheet Standing Wave

We are going to look at a continuous stream of waves produced on a string by a vibrating electromagnet. The waves produced on the string are traveling to the right and can be described by the equation

If we take the origin to be the location where the waves are generated, this equation gives the height of the string y1 at time t and at a distance x from the origin. ym is the maximum

amplitude of the string, k is a constant related to the wavelength ( , it is called the

wave number and measured in radians per meter) and  is a constant related to the period ( , it is called the angular frequency and measured in radians per second).

When these waves reach the end of the string, they hit a boundary and are reflected back. The reflected waves are traveling to the left and they can be described by the equation

The height of the string at any location along the string is found by adding the heights of the two individual waves.

1. Find an equation for the string’s height

y = y1 + y2

After a bit of algebra, you should get

This is the equation of a standing wave. It has an amplitude of 2ym. It is called a standing wave because the position and time dependence have been separated. The amplitude of the simple harmonic motion of any particle of the string varies according to its position. If we let , this equation looks like the equation of a regular cosine function

For this equation we call A the amplitude and the value of y varies according to the value of cos t. So for the standing wave equation, the quantity , is the maximum amplitude of the string’s displacement at position x. The height of the string at any location will oscillate up and down as this amplitude is multiplied by cos t.

In addition to standing waves occurring on strings, a standing wave can also occur with light waves (in lasers) and with sound waves (in organ pipes). The distinctive features of a standing wave are:

 There is no sense of motion. A standing wave is an oscillation pattern with a stationary outline.

 Nodes are locations where a standing wave has zero displacement.

 Antinodes are locations where a standing wave has its maximum displacement.

 Every particle of the medium oscillates in Simple Harmonic Motion (back and forth) with the same frequency .

 The amplitude of motion depends on the location x of the particle in the medium.

 A given particle in a standing wave vibrates within an envelope function 2ym sin kx. This is in contrast to a traveling wave where all particles oscillate with the same amplitude.

For a vibrating string, the locations where the string does not move are called nodes. A string can vibrate with different numbers of nodes depending on the tension in the string. If nodes occur just at the ends of the string, the string is said to be vibrating in its

If the string is vibrating with three nodes, the string is said to be vibrating at its second harmonic with the wavelength given by

If the string is vibrating with four nodes, it is said to be vibrating at its third harmonic, and the wavelength is

In general, a string vibrates with n+1 nodes, and we can determine the wavelength that occurs when a string vibrates with different harmonics from the equation

n = 1, 2, 3, …

3. For the standing wave that you are currently observing, what is the wavelength?

4. Find the locations (x values) of all of the nodes of this standing wave.

Antinodes are positions at which the maximum displacement occurs.

5. From the equation for a standing wave, , find the locations of the antinodes in terms of the wavelength ().

Instructor Notes:

Learning Outcomes:

Upon completion of this module the students should be able to:

 Use the sum and difference formulas to simplify a trigonometric function and

 Describe the difference between a standing wave and a travelling wave.

Equipment: variety of masses, string, C-clamp, pulley, yardstick and oscillator with a metal bar. Have the distance from the oscillator and pulley about 1.06 meter (97.5 cm from edge of oscillator to edge of pulley for yellow string) with about 70 g on the end of the string.

3. L =1.5 m n = 4 4. Five nodes:

ith _{node: i = 1, 2, 3, 4, 5}

x1 = 0 m x4 = 0.8625 m

x2= 0.2874 m x5 = 1.15 m

x3= 0.575 m

5. Maximum values occur when sin(kx) = 1

Since

In general, where xj is the jth _{antinode and j = 1, 2, 3, 4, …, n}

6. Four antinodes

jth _{node: i = 1, 2, 3, 4}

x1 = 0.14375 m x3 = 0.71875 m

x2= 0.43125 m x4 = 1.00625 m

With the velocity v given by , where T is the tension in the string and  is the linear mass density, the natural frequencies of the string are:

n = 1, 2, 3,