A Level Physics - Stationary Waves

How are standing waves produced?

#Standing waves are produced by the superposition of two waves of the same frequency and amplitude travelling in opposite directions

What does a stationary waves store?

#Energy instead of transferring it like a progressive wave

What are the features of progressive waves?

#

• All points have the same amplitude (in turn)

• Points exactly a wavelength are in phase. The phase of points within one wavelength can be 0 to 360

• Energy is transferred along the wave

• Does not habe nodes or antinodes

• The wave speed is the speed at which the waves moves through a medium 

What are the features of stationary waves?

• #

• Each point has a different aplitude depending on the amount of superposition

• Points between nodes are in phase. Points on either side of a node are out of phase

• Energy is stored, ot transferred

• Has nodes and antinodes

• Each point on the wave oscillates at a different spedd. The overall doesn’t move

What are stationary waves made up of?

#nodes and antinodes

What are nodes?

#Regions where there is no vibration

What are antinodes?

#Regions where the vibrations are at their maximum amplitude

What can the nodes and antinodes not do?

#Nodes and antinodes do not move along the string

• Nodes are fixed and antinodes only move in the vertical direction

What is the phase difference between two points on a stationary wave?

#The phase difference between two points on a stationary wave are either in phase or out of phase

What is the phrase to describe the distance between nodes ?

• Points between nodes are in phase with each other

What does the principle of superposition state_

#When two or more waves with the same frequency arrive at a point, the resultant displacement is the sum of the displacements of each wave

What happens when two waves with the same frequencz and amplitude arrive at a point in phase?

• #In phase, causing constructive interference. The peaks and troughs line up on both waves and the resultant wave has double the amplitude

What happens when two waves with the same frequencz and amplitude arrive at a point anti phase?

#In anti-phase, causing destructive interference. The peaks on one wave line up with the troughs of the other. The resultant wave has no amplitude

How do stationary waves form?

#Two waves travelling in opposite directions along the same line with the same frequency superpose

What are the requirements to form a stationary wave?

#The waves must have:

• The same wavelength)

• A similar amplitude

What occurs at the nodes of stationary wave?

#The waves are in anti-phase meaning destructive interference occurs

• This causes the two waves to cancel each other out

What occurs at the antinodes of stationary wave?

#The waves are in phase meaning constructive interference occurs

• This causes the waves to add together

Examples of stationary waves

#Stationary Microwaves

#Stationary sound waves

How are stationary microwaves formed?

• #

• A microwave source is placed in line with a reflecting plate and a small detector between the two

• The reflector can be moved to and from the source to vary the stationary wave pattern formed

• By moving the detector, it can pick up the minima (nodes) and maxima (antinodes) of the stationary wave pattern

What happens as you change the frequency of a standing wave?

#As the frequency is increased, more harmonics begin to appear

What is the first harmonic or fundamental frequency?

#The simplest wave pattern is a single loop made up of two nodes and an antinode

For a string of length L, what is the frequency?

#For a string of length L, the wavelength of the lowest harmonic is 2L

• This is because there is only one loop of the stationary wave, which is a half wavelength

How s the frequency in first harmonic calculated?

#V/2L

What are the features of the second harmonic?

• #

• The second harmonic has three nodes and two antinodes

• The wavelength is L

How is the frequency of the second harmonic calculated?

V /L

What are the features of third harmonic?

• #

• The third harmonic has four nodes and three antinodes

• The wavelength is 2L / 3

How is the frequency of the third harmonic calculated?

3v/2L

f3 = 3f1

What is the general formula for the number of nodes and anti-nodes of nth harmonic ?

#The nth harmonic has n antinodes and n + 1 nodes

What is the frequency of the nth harmonic?

#nth harmonic the frequency = n × frequency of first harmonic

What is the equation for the speed of a wave travelling along a string with two fixed ends?

#v = (T/U)^(1/2)

• Where:

  • T = tension in the string (N)

  • μ = mass per unit length of the string (kg m^–1)

What is the equation for the first harmonic?

#f = (1/2L) x (T/U)^(1/2)

• Where:

  • f = frequency (Hz)

  • L = the length of the string (m)

What is the independent variable for Required Practical: Investigating Stationary Waves?

#either length, tension, or mass per unit length

What is the dependent variable for Required Practical: Investigating Stationary Waves?

#frequency of the first harmonic

What is the control variable for Required Practical: Investigating Stationary Waves?

#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

Add Equipment for Required practical

#test

What are the resolutions of meauring equipment?

#Resolution of measuring equipment:

• Metre ruler = 1 mm

• Signal generator ~ 10 nHz

• Top-pan balance = 0.005 g

What is the method for Required practical(stationary waves)

#

1. 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 attach the mass hanger

2. Adjust the position of the bridge so that the length L is measured from the vibration generator to the bridge using a metre ruler

3. Turn on the signal generator to set the string oscillating

4. Increase the frequency of the vibration generator until the first harmonic is observed and read the frequency that this occurs at

5. Repeat the procedure with different lengths

6. Repeat the frequency readings at least two more times and take the average of these measurements

7. Measure the tension in the string using T = mg

   • Where m is the amount of mass attached to the string and g is the gravitational field strength on Earth (9.81 N kg^–1)

8. 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

How can you use results of stationary waves experiment to create line graph?

#V= fλ = f × 2L

• Rearranging for frequency, f:

• F =(v/2)(1/L)

• 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)

1. Plot a graph of the mean values of f against 1/L

2. Draw a line of best fit and calculate the gradient

3. Work out the wave speed, which will be 2 × gradient:

How can you reduce systematic errors in Standing waves practical?

• #

• 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

How can you reduce random errors in Standing waves practical?

• #

• 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

• 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 the frequency of this

  • If taking three repeat readings, repeat this procedure again

  • Average the three readings and move onto the next measurement

How can repeat measurments for frequency be best measured?

#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 the frequency of this

• If taking three repeat readings, repeat this procedure again

• Average the three readings and move onto the next measurement

What are safety considerations for standing waves practical

• #

• 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 in case it snaps

• Stand well away from the masses in case they fall onto the floor

• Place a crash mat or any soft surface under the masses to break their fall