# Standing Waves

## Standing Wave Patterns

#### Purpose

To produce standing wave patterns along a string.

#### Objectives

• To know how to use the concept of linear mass density.
• To be able to calculate the speed of waves traveling along a stretched string, and the wavelength of the waves.
• To measure the distance between the nodes of a standing wave pattern and determine its relationship to the wavelength.
• To learn what factors determine the conditions for standing wave patterns along a string.
• To be able to produce standing wave patterns along a string by varying the length of the string.
• To learn how record and organize experimental data using spreadsheets.

#### Equipment

• Oscillator and a string
• Hooks with mass weights
• Pasco Interface and a computer

#### Theory

A wave with frequency (f) and wavelength ( λ) travels with velocity given by:

velocity  =  (frequency)  ×  (wavelength)      (Eq. 1)

If that wave travels along a string, the velocity must also be equal to

velocity   =    (Tension) ⁄ (linear mass density)   (Eq. 2)

The linear mass density (μ) is the mass of the string per unit length, or:

μ   =   (mass) ⁄ (length)     (Eq. 3)

When a wave travels along a string and reaches its end, part of the energy is reflected back along the string. Given the right conditions that depend on the tension, the mass, and the length of the string, a standing wave pattern occurs. The string then vibrates with large amplitudes at some points (antinodes) and it does not vibrate at all at other points (nodes). A pattern of “loops” appear along the string, with each “loop” stretching from a node to a node.

#### Experiment, Data and Results

##### Preliminary Setup
• Slide a mechanical oscillator onto a rod, secure it tightly so that the string that goes over the pulley to the hook is horizontal. Use two wires to connect the oscillator to the PASCO PowerAmplified that connects to the PASCO Interface. Open DataStudio, select “New Experiment”, Click and Drag “Power Amplified” to one of the analog intputs. Select “sin” form with frequency 25 Hz and amplitude of 5V in the control panel.

• Add 150 g to the 50-g hook and hang them on the loose end of the string over the table pulley. Calculate the tension in the string:

Tension  = (mass) ×(g)  =

• Note on units and conversion. If you work in [kg] and [m], you may use 9.8 m/s2 for the gravitational acceleration. If you work in [g] and [cm], the gravitational acceleration is 980 cm/s2.
• Calculate the velocity of the waves along the string using Eq. 2. The linear mass density (μ) is 0.011 g/cm or 0.0011 kg/m, depending on which system of units you have determined to use.

Velocity   =   √ (Tension) ⁄ (linear mass density)  =

• Calculate the wavelength of the wave using Eq. 1:

(wavelength)   = (velocity) ⁄ (frequency)   =

#### Questions.

If you double the frequency but keeping the tension and the linear mass density the same, what happens to the wavelength? What happens if you triple the frequency? Fill in the table below for the wavelength:

 frequency [Hz] wavelength [m]

#### Activity

Turn on the power. Adjust the length of the string by moving the vertical rod until a standing wave pattern is established.

Measure the distance between two nodes.

Repeat for 50 Hz, 75 Hz, 100 Hz. Write downs your results in a table

 frequency [Hz] wavelength [m] distance between nodes [m]

#### Questions.

What is the relationship between the wavelength of the waves producing the standing wave pattern and the distance between two adjacent nodes?

What frequency will produce the next standing wave pattern after 100 Hz?

If a wave with frequency 60 Hz produces 1-loop standing wave pattern, with what frequency can you produce 3 loops?

If a wave with frequency 60 Hz produces 3-loop standing wave pattern, with what frequency can you produce 4 loops?