3. Superposition + Standing Waves

What is the principle of superposition?

When two waves meet at a point, the resultant displacement is the sum of the two individual displacements

What is constructive + destructive interference?

• Constructive interference - displacements add to give a greater displacement

• Destructive interference - displacements add to give a smaller displacement

What does path difference mean?

Difference in how far two waves have travelled to a certain point

When does constructive + destructive displacement occur?

Constructive:

• In phase - phase difference of a multiple of 2π (e.g 0, 2π, 4π)

• Path difference is whole number of wavelengths

Destructive

• Antiphase - phase difference of an odd multiple of π (e.g π, 3π, 5π)

• Path difference is a half number of wavelengths

What does coherent mean + condition needed?

• Coherent - constant phase difference between two waves

• Condition needed - same frequency

What is two-source interference + condition needed?

• Two-source interference - waves from two source interfere to produce an interference pattern of maximas + minimas

• Condition needed - two coherent sources

Where do maximas + minimas occur + order?

• Maxima - when path difference is a whole number of wavelengths → constructive interference

• Minima - when path difference is half a wavelength → destructive interference

Order:

1. Central maxima

2. 1st order minima

3. 1st order maxima

4. 2nd order minima etc

How to investigate two source interference of sound + microwaves?

Sound:

1. Setup amplifier connected to two speakers

2. Walk parallel to line of speakers

3. There will be areas of loud (constructive interference) + quiet sound (destructive interference)

Microwaves:

1. Setup signal generator connected to two microwave transmitter cones

2. Move microwave receiver parallel to line of cones

3. There will be areas of strong (constructive interference) + weak signals (destructive interference)

How to do Young’s double slit experiment?

1. Setup a mounted card with two slits the same size as the wavelength of the laser light

2. Shine a laser through the slits

3. An interference pattern of light (maxima) + dark fringes (minima) should form

What is the equation for the double-slit experiment?

λ = ax / D (wavelength = spacing between slits x fringe spacing between two adjacent maxima / distance between slit + screen)

What three conditions are needed for stationary waves to form?

Two waves:

• with same frequency

• with same amplitude

• travelling in opposite directions

What is a node + antinode?

• Node - the two waves meet in antiphase → amplitude is always zero

• Antinode - the two waves meet in phase → amplitude can reach maximum

How to find the wavelength of stationary wave?

Distance between two nodes x 2

What are three differences between progressive + stationary waves?

• Progressive waves transfer energy whereas stationary waves store energy

• Progressive waves have the same amplitude in all parts whereas stationary waves have varying amplitude throughout

• Progressive waves have continuously varying phase difference whereas all point between two nodes in a stationary waves are in phase + points in adjacent nodes are in antiphase

What do fundamental mode of vibration + fundamental frequency mean?

• Fundamental mode of vibration (1st harmonic) - one half wavelength fits on string → wavelength is 2L

• Fundamental frequency (f₀) - frequency of this wave

What is the equation for the wavelength + frequency of the nth harmonic?

• λ = 2L / n (wavelength = 2 x length of string / n)

• f = nf₀ (frequency = n x fundamental frequency)

How are stationary waves formed in stringed instruments?

1. Your finger creates a node at the point where you press the string

2. The string vibrates

3. Waves are sent out at both directions + reflected → stationary wave

How are stationary waves formed in a closed end tube + wavelength + frequency?

• Node at closed end

• Antinode at open end

• λ = 4L / 2n - 1 (wavelength = 4 x length of string / 2n - 1)

• f = (2n - 1) f₀ (frequency = (2n - 1) x fundamental frequency)

How are stationary waves formed in a open end tube + wavelength + frequency?

• Antinode at both ends

• λ = 2L / n (wavelength = 2 x length of string / n)

• f = nf₀ (frequency = n x fundamental frequency)

How to investigate stationary waves using microwaves?

1. Setup a microwave transmitter pointed at a metal plate

2. Move a microwave receiver connected to a speaker perpendicular to the transmitter

3. Max sound → antinode

4. Min sound → node

How to find the speed of sound using stationary waves?

1. Setup a hollow tube filed with water in a measuring cylinder

2. Choose a tuning fork + record the frequency of it

3. Hit the tuning fork on a table + hold it above the tube

4. Move the tuning fork up + down until the sound is the loudest

5. Measure distance between top of water to tuning fork

6. Repeats + average

7. Repeat for other tuning forks with different frequencies

8. Plot a graph of frequency (y-axis) against 1 / wavelength (x-axis)

9. The gradient is the speed of sound