# Standing Waves

#### Purpose

To produce standing wave patterns along a string.

#### Objectives

• To be able to create standing waves along a string.
• To determine the wavelength of the waves creating the standing waves.
• To learn how to record and organize experimental data using spreadsheets.
• To learn to identify factors that affect the precision of experimental measurements.

#### Equipment

• Oscillator with a string
• Vertical Rod, hooks with mass weights, electric wires
• Pasco Interface and a computer

#### Theory

The velocity of transverse waves along a string with tension $\tau$ and linear mass density $\mu$ is given by

$v = \sqrt{\frac{\tau}{\mu}}$

When a standing wave pattern is created, the distance from a node to node (the length of one loop) is equal to half of the wavelength of the waves.  Thus for n number of loops, the distance over which the standing wave spans and the wavelength are related as:

$n \frac{\lambda}{2} = d$,

where $n$ is the number of loops, $d$ is the distance of the entire standing wave pattern, and $\lambda$ is the wavelength of the wave.

#### Experimental Setup

Here are step-by-step instructions on how to set up the lab experiment.

Hang total of 200 g  (hook plus weights) to the string and calculate the tension, $\tau = mg$.

The linear mass density of the string is 0.0011 kg/m (pale pink) or 0.0014 kg/m (bright pink). Calculate the velocity of the waves along the string.

Follow the Capstone Instructions for Signal Generator to set up the computer.

#### Data

Set the generator to 25 Hz and 5 V amplitude. Create a single-loop standing wave. Measure the distance over which the loop spans and calculate the wavelength of the wave. Repeat for 50, 75, and 100 Hz. Write down your results in a table.

Set up the following data table in a spreadsheet and record your measurements. Use the spreadsheet built-in function for calculations and not your calculators.

 f (Hz) Loops d (m) wavelength (m) F(Hz) % Diff. 25 1

#### FAQ

##### What is the distance d?

The distance d is measured for the entire standing wave pattern. For example, if you see a 3-loop pattern, measure how far the three loops stretch and record it as the distance d in meters.

##### How do we calculate the wavelength?

Each loop stretches $\latex \lambda/2$ distance. So, if you measure a the distance for a 3-loop pattern to be 0.87 m, then you calculate the wavelength by using the following formula:

$(3) \times \frac{\lambda}{2} = d$

##### How do we calculate the percent difference in the last column?

The first column is the frequency set through the generator. The second to last column is the frequency calculated based on your measurements ($f = v / \lambda$). The last column is the percent difference between the two:

$\% = \frac{f_1-f_2}{f_{av}}$

#### Results

You should get percent difference between the frequencies from the computer and what you measure in the range of 1-5 %. If you have larger discrepancy, you must repeat your measurements and identify the factors that are causing that discrepancy.