### Core Practical 7: Investigating Stationary Waves

#### Aims of the Experiment

- The overall aim of the experiment is to measure how the frequency of the first harmonic is affected by changing one of the following variables:
- The length of the string
- The tension in the string
- Strings with different values of mass per unit length

#### Variables

- Independent variable = either length, tension, or mass per unit length
- Dependent variable = frequency of the first harmonic
- Control variables
- If length is varied = same masses attached (tension), same string (mass per unit length)
- If tension is varied = same length of the string, same string (mass per unit length)
- If mass per unit length is varied = same masses attached (tension), same length of the string

#### Equipment List

- Resolution of measuring equipment:
- Metre ruler = 1 mm
- Signal generator ~ 10 nHz
- Top-pan balance = 0.005 g

#### Method

*The setup of apparatus required to measure the frequency of the first harmonic at different values of length, tension, or mass per unit length*

This method is an example of the procedure for varying the length of the string with the frequency – this is just one possible relationship that can be tested

- Set up the apparatus by attaching one end of the string to the
**vibration generator**and pass the other end over the bench pulley and secure to the mass hanger - Adjust the position of the bridge so that the length
*L*is measured from the vibration generator to the bridge using a metre ruler - Turn on the
**signal generator**to set the string**oscillating** - Increase the
**frequency**of the vibration generator until the**first harmonic**(**nodes**at both ends and an**antinode**in the middle) is observed and read the frequency that this occurs at **Repeat**the procedure with different lengths of*L*- Repeat the frequency readings at least two more times and take the
**average**of these measurements - Measure the
**tension**in the string using*T*=*mg*- Where
*m*is the mass attached to the string and*g*is the gravitational field strength on Earth (9.81 N kg^{–1})

- Where
- Measure the
**mass per unit length**of the string,*μ*= mass of string ÷ length of string- Simply take a known length of the string (1 m is ideal) and measure its mass on a balance

- An example of a table with some possible string lengths might look like this:

*Before conducting an experiment, a table must be set up to detail of what measurements are to be made*

#### Analysing the Results

- For the
**first harmonic**, wavelength,*λ*= 2*L* - So, the
**speed**of the stationary wave is:

*v* = *f*λ = *f* × 2*L*

- Rearranging for
**frequency**,*f*:

- Comparing this to the equation of a straight line: y = mx
- y =
*f*(Hz) - x = 1/
*L*(m^{–1}) - Gradient =
*v*/2 (m s^{–1})

- y =

**Plot**a graph of the mean values of*f*against 1/*L*- Draw a
**line of best fit**and calculate the gradient - Work out the
**wave speed**, which will be 2 × gradient

**If the frequency is plotted against the inverse of the length, the velocity is twice the gradient of the graph**

- Verify the
**wave speed**of the travelling waves using the equation:

- Where:
*T*= tension (N)*μ*= mass per unit length (kg m^{−1})

- Assess the
**uncertainties**in the measurements of length and frequency, and carry out calculations to determine the uncertainty in the wave speed

#### Evaluating the Experiment

**Systematic errors:**

- An
**oscilloscope**can be used to verify the signal generator’s readings - The signal generator should be left for about 20 minutes to stabilise
- The measurements would have a
**greater resolution**if the length used is as large as possible, or as many half-wavelengths as possible- This means measurements should span a
**suitable range**, for example, 20 cm intervals over at least 1.0 m

- This means measurements should span a

**Random errors:**

- The
**sharpness**of resonance leads to the biggest problem in deciding when the first harmonic is achieved- This can be
**resolved**by adjusting the frequency while looking closely at a node. This is a technique to gain the largest response - Looking at the amplitude is likely to be less reliable since the wave will be moving very fast

- This can be
- When taking repeat measurements of the frequency, the best procedure is as follows:
- Determine the
**frequency**of the**first harmonic**when the largest vibration is observed and note down the frequency at this point **Increase the frequency**and then gradually reduce it until the first harmonic is observed again and note down this frequency- If taking three repeat readings, repeat this procedure again
- Average the three readings and move on to the next measurement

- Determine the

#### Safety Considerations

- Use a
**rubber string**instead of a metal wire, in case it snaps under tension - If using a metal wire,
**wear goggles**to protect the eyes **Stand well away**from the masses in case they fall onto the floor- Place a
**crash mat**or a soft surface under the masses to break their fall

#### Worked Example

A student investigates the relationship between the frequency of the stationary waves on a wire and the tension in the wire. The tension is varied by adding masses to a hanger, which is attached to a pulley over one end of a table. The student records the following data:

- Mass of the wire,
*m*= 0.16 g - Length of the wire weighed,
*l*= 1.0 m - Distance between the fixed ends,
*L*= 0.4 m

Calculate the frequency using the relation between frequency, length, tension and mass per unit length. Evaluate the percentage uncertainty in these values.