Interference of Waves

1. Interference of waves occurs when 2 sets of continuous waves meet and overlap and their wavefronts superimpose each other in accordance with the principle of superimposition.

2. Principle of superimposition states that: when 2 waves overlap, the resultant displacement is equal to the sum of displacement of the 2 individual waves.

3. Effects of Interference:

• Change in amplitude:

• Constructive interference produces maximum amplitude (max. crest or trough) at points of antinodes; Lines that join points of antinodes are known as antinodal lines.

• Destructive interference produces zero amplitude at points of nodes. Lines that join nodes are known as nodal lines. 

• No change in: Frequency ( f), wavelength (λ) and speed (v) IF the waves are from coherent wave sources:
• Coherent waves are waves which maintain a constant phase difference and can be produced by 2 oscillating sources vibrating at the same frequency.

4. Overlapping waves interfere either constructively or destructively:

• Constructive Interference: When 2 waves meet with the same amplitude in the same direction (e.g. crest meets crest or trough meets trough), the resultant displacement is the combination of the 2 amplitudes in the same direction (e.g. higher crest or deeper trough of double amplitude) - this is known as constructive interference.

• Destructive Interference: When 2 waves meet with the same amplitude in opposite directions, they cancel each other out - the resultant displacement is zero - this is known as destructive interference.

5. Experiments Showing Interference in Waves:

• Water Waves - Ripple Tank & 2 Coherent Water Wave Sources (Fig. 1.31 at pg. 21 of F5 textbook) (SPM 2009 P3, Q2 at pg. 224) / (2011 P3, Q4 at pg. 328);
• Light Waves - Thomas Young's "Double-Slit" Experiment (Fig. 1.32 at pg. 21 of F5 textbook)(SPM 2010 P3 Q2 at pg. 273) / 

          (SPM 2010 P2 Q10(a) at pg 260) /
          (SPM Yr 2012 P1 Q33 @ pg 339);

• Sound Waves - Common Audio Signal to 2 Loud Speakers Experiment (Fig. 1.33 at pg. 22 of F5 textbook) (SPM 2009 P2 QA6 at pg 205) / (2010 P2 Q6 at pg. 253)

6. Approximate Formula for Wavelength (λ = ax/D) from Interference Pattern:

• where, λ = wavelength

                   a = distance between the 2 coherent wave sources
               
                   x = distance between 2 adjacent antinodal lines
                         (radiating ripples or interference fringes or points of loudness)

                   D = perpendicular distance between the parallel lines
                          where a and x are  measured 

• From experiment, empirical evidence shows:
• that x is directly proportional to λ
• that x is inversely proportional to a
• therefore, mathematically, x = kλ/a, where k is found to be = D
• Thus, x = Dλ/a
• And, mathematically, λ = ax/D

• Thus, the "Approximate Formula for Wavelength" can be used to find the wavelength of waves from their interference patterns. 

• From the formula λ = ax/D or x = Dλ/a (same formula with different subjects)
• Interference pattern of red, green or blue light can all be explained by the "Wavelength Approximate Formula, x = Dλ/a" 
• The distance between adjacent fringes x of red light is bigger than that of green or blue because the wavelength of red light (λ _red) is longer as compared to that of green light (λ_green) or blue light.
• Since  λ _red > λ_green > λ_blue, from  x = Dλ/a, therefore, x _red > x_green > x_blue.

• Application of Interference:
• In water waves - Spherical bow of ship produces destructive interferences to lessen resistance to movement of ship thereby saving energy;
• In sound waves - Destructive interference is used to silent engine noise in cabin of aeroplane, car and headphone (please see below)
• In light waves - Interference pattern and the "Wavelength Approximate Formula" is used to find the wavelength of waves.

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       Application of Destructive Interferenceof Sound Waves

Noise Cancellation

• Noise cancellation is a method to reduce or completely cancel out undesirable sound.
• A noise-cancellation speaker emits a sound wave of equal but opposite amplitude and same frequency with the original sound i.e. by emission of anti-phase sound waves.
• The sound waves will overlap each other in a process called destructive interference, causing the waves to cancel each other out and there would be no sound.
• The sum of the amplitudes of the waves is equal to zero.

 

Application of noise cancellation:

1. Headphone - people working near aircraft or in noisy factories can now wear these electronic noise cancellation headsets to protect their hearing.
2. Cars - The way it works is that a microphone connected to the car stereo system picks up all the sound inside the car, including music or such from the stereo. Then the noise-cancellation system produces noise-canceling sound waves that match the frequency of unwanted sound.
3. Aircraft - The system uses microphones to pick up the vibrations due to jet's engine in the cabin walls. It then analyzes the signals and generates counter vibrations in the walls to produce a net result of zero vibrations.