Resonance and Huygens’ Principle

Resonance! When an oscillating system is being pushed by an oscillating force of constant frequency, the system can undergo resonance under certain specific conditions. Some necessary terms Driving force – a force that is oscillating in magnitude, pushing the system at constant frequency (think of pushing a person on a swing) Natural frequency – the frequency that the oscillating system normally has, in the absence of a driving force (think of the frequency that the swinging person would have if they were not being pushed at all) Driving frequency – the frequency of the oscillating force that is pushing the system (think of the frequency of the person’s push)

When does resonance occur? Think of pushing a person on a swing – When would you push in order to maximize their amplitude? Every time the person returns to the same point! If the frequency of your push matches the natural frequency of the swinging person, they will go very high! Resonance occur when the frequency of the driving force matches the natural frequency of the system.

Resonance whiteboard You are hanging a 2-kg block from a spring of constant 300 N/m. If you were to oscillate your hand up and down, with what driving frequency would you achieve the largest amplitude for the block?

Solution ≈ 0.5 s The natural frequency of the system can be found by using f natural = 1/T f natural ≈ 2 Hz If you oscillate the system with a driving frequency of about 2 Hz, the block will obtain a very large amplitude!

When the driving frequency matches the natural frequency, resonance!!! This results in some truly amazing phenomena. If you match the resonance frequency of a wineglass, you can shatter it with sound! Soldiers are required to break step (stop marching) while crossing over a bridge. Why?

Christiaan Huygens ( ) Theorized an ingenious model of wave propagation that still holds true today. Its applications range studies of sound to light, and even quantum mechanics. “When a wave propagates through a medium, each point of disturbance created by that wave behaves itself as a source of new waves.”

Hugyens was first prompted to study the propagation of waves by observing water waves going toward a small opening in a barrier. The observation was quite mesmerizing, as well as surprising – and was unexplained by contemporary wave theory at the time. Plane waves in water going toward a wall with a small opening (lines represent crests) Surprising results!

Huygens’ Principle When a wave propagates through a medium, each point on that wave acts as a source of new “wavelets”, each of which travel outward in all directions. It is the superposition of all of these wavelets that determines the subsequent development of the wave. Each point on the wave acts as a source of new waves! It is the superposition of all of these “wavelets” that governs the behavior of the wave.

Applied to a spherical wave When Huygens’ Principle is applied to a spherical wave, the results are consistent with the observed “spreading out” of the concentric wavefronts.

When all of the “wavelets” are added together using the Principle of Superposition, the result is familiar! Net wave

Applied to a Plane Wave! Net result, using the Principle of Superposition

Amazing! This theory was consistent with Huygens’ observations, and predicts that the center point of the wavefront, when it reaches the hole in the wall, behaves as a source of waves into the other side of the wall, whereas all other parts of the wave do not affect the other side. This is what is looks like!

Whiteboard Wavelets! There is a sound-proof wall in between you and a speaker that is playing a constant frequency. Will you hear anything? Use Huygens’ Principle to support your answer.

Huygens’ Principle Summary When a wave propagates through a medium, it can be modeled as each point on the wave creating new waves. This is a useful mathematical and conceptual tool that explains some surprising phenomena. We will use it thoroughly to explore double-slit and single-slit interference! – Keep Huygens in mind!

Two-Source Interference When two sources of waves are close to one another, they create a beautiful and complex interference pattern. This can be understood by using the Principle of Superposition! Principle of Superposition 1)When the crest of one wave meets the crest of another, or the trough of one meets the trough of another, they will constructively interfere and create a large combined wave. 2)When the crest of one wave meets the trough of another, they will destructively interfere, negative one another.

Two Synched Sources of Sound! Let’s consider two speakers that are in phase with one another. If two sources are in phase, it means that they emit crests and troughs at the same time as one another. They are synched up! When one is emitting a crest, the other is also emitting a crest, etc.

Constructive Interference Consider the two in-phase sources of sound shown below. Source 1 Source 2 Take the point X to be a distance L 1 away from Source 1, and a distance L 2 away from Source 2. L1L1 L2L2 This means that by the time the waves reach X, waves from Source 1 have traveled a distance L 1, and waves from Source 2 have traveled a distance L 2.

If L 1 = L 2 … Source 1 Source 2 If wave 1 has undergone a certain number of full oscillations by the time it reaches X… Then wave 2 has also undergone the same number of full oscillations by the time it reaches X! Since the waves were in phase when they were emitted, and traveled the same distance, they will still be in phase when they meet at X! X will be a loud spot of constructive interference!!

What if we choose a point such that L 1 is exactly one wavelength greater than L 2 ? Source 1 Source 2 Wave 1 has undergone a certain number of full oscillations by the time it reaches X Wave 2 has only undergone 1 less oscillation by the time it reaches X! Even though Wave 1 has undergone one full oscillation more than Wave 2, the waves are still synched up when they interfere at point X! This will also be a loud spot (constructive interference).

Constructive interference will occur if L 1 – L 2 = 0 L 1 – L 2 = λ (Wave 1 has traveled a full extra wavelength by the time that they meet) L 2 – L 1 = λ (Wave 2 has traveled a full extra wavelength by the time that they meet) L 1 – L 2 = 2λ (Wave 1 has traveled two extra wavelengths by the time that they meet) L 2 – L 1 = 2λ (Wave 2 has traveled two extra wavelengths by the time that they meet)

In general, Source 1 Source 2 L1L1 L2L2 Constructive interference will occur if m = 0 if the waves have traveled the same distance, m = 1 if one of the waves has traveled one extra wavelength, m = 2 if one of the waves have traveled two extra wavelengths, etc.

Interference

Two-Source Interference When two sources of waves are close to one another, they create a beautiful and complex interference pattern. This can be understood by using the Principle of Superposition! Principle of Superposition 1)When the crest of one wave meets the crest of another, or the trough of one meets the trough of another, they will constructively interfere and create a large combined wave. 2)When the crest of one wave meets the trough of another, they will destructively interfere, negative one another.

Two Synched Sources of Sound! Let’s consider two speakers that are in phase with one another. If two sources are in phase, it means that they emit crests and troughs at the same time as one another. They are synched up! When one is emitting a crest, the other is also emitting a crest, etc.

Constructive Interference Consider the two in-phase sources of sound shown below. Source 1 Source 2 Point X is a distance L 1 away from Source 1, and a distance L 2 away from Source 2. L1L1 L2L2 This means that by the time the waves reach X, waves from Source 1 have traveled a distance L 1, and waves from Source 2 have traveled a distance L 2.

If L 1 = L 2 … Source 1 Source 2 If wave 1 has undergone a certain number of full oscillations by the time it reaches X… Then wave 2 has also undergone the same number of full oscillations by the time it reaches X! Since the waves were in phase when they were emitted, and traveled the same distance, they will still be in phase when they meet at X! X will be a loud spot of constructive interference!!

What if we choose a point such that L 1 is exactly one wavelength greater than L 2 ? Source 1 Source 2 Wave 1 has undergone a certain number of full oscillations by the time it reaches X Wave 2 has only undergone 1 less oscillation by the time it reaches X! Even though Wave 1 has undergone one full oscillation more than Wave 2, the waves are still synched up when they interfere at point X! This will also be a loud spot (constructive interference).

Constructive interference will occur if L 1 – L 2 = 0 L 1 – L 2 = λ (Wave 1 has traveled a full extra wavelength by the time that they meet) L 2 – L 1 = λ (Wave 2 has traveled a full extra wavelength by the time that they meet) L 1 – L 2 = 2λ (Wave 1 has traveled two extra wavelengths by the time that they meet) L 2 – L 1 = 2λ (Wave 2 has traveled two extra wavelengths by the time that they meet)

In general, Source 1 Source 2 L1L1 L2L2 Constructive interference will occur if m = 0 if the waves have traveled the same distance, m = 1 if one of the waves has traveled one extra wavelength, m = 2 if one of the waves have traveled two extra wavelengths, etc.

Constructive interference will occur along all of these lines (where crests from source 1 meet crests from source 2) The central line is the m = 0 line. This means that waves from both sources will have traveled the same distance by the time that they reach any point on this line. L 1 = L 2 m = 0

The lines of constructive interference adjacent to the center line are the m = 1 lines. This means that waves from one of the sources will have traveled exactly one wavelength further than waves from the other source by the time that they reach any point on these lines. m = 1

The next lines of constructive interference are the m = 2 lines. This means that waves from one of the sources will have traveled exactly two wavelengths further than waves from the other source by the time that they reach any point on these lines. m = 2

m = 0 Constructive interference will occur on any of these lines, because they satisfy the condition m = 1 m = 2

Lab Challenge: Two speakers are in-phase, and placed at one end of the classroom. Determine the frequency produced by the speakers by using the sound interference pattern and a measuring tape. Put your experimental procedure (diagram) and your calculations/results on a whiteboard!

Destructive Interference Consider the two in-phase sources of sound shown below. Source 1 Source 2 The point X is a distance L 1 away from Source 1, and a distance L 2 away from Source 2. L1L1 L2L2 This means that by the time the waves reach X, waves from Source 1 have traveled a distance L 1, and waves from Source 2 have traveled a distance L 2.

If L 1 is exactly one half wavelength less than L 2 … Source 1 Source 2 A crest from one wave will meet a trough from the other! The waves will destructively interfere. 2.5 oscillations 3 oscillations

Destructive interference will occur if L 1 – L 2 = λ/2 (Wave 1 has traveled an extra half-wavelength by the time that they meet) L 2 – L 1 = λ/2 (Wave 2 has traveled an extra half-wavelength by the time that they meet) L 1 – L 2 = 3λ/2 (Wave 1 has traveled an extra 1.5 wavelengths by the time that they meet) L 2 – L 1 = 3λ/2 (Wave 2 has traveled an extra 1.5 wavelengths by the time that they meet)

In general, Source 1 Source 2 L1L1 L2L2 Destructive interference will occur if m = 1 if one wave has traveled an extra half-wavelength, m = 2 if one wave has traveled an extra 1.5 wavelengths, etc.

Destructive interference will occur along these lines (where crests from source 1 meet troughs from source 2) The first line of destructive interference is m = 1. This means that waves from one source have traveled λ/2 further than waves from the other source by the time that they reach any point on this line. m = 1

The second line of destructive interference is m = 2. This means that waves from one source have traveled 1.5 λ further than waves from the other source by the time that they reach any point on this line. m = 2

Destructive interference will occur on any of these lines, because they satisfy the condition m = 1 m = 2 m = 3

Three Stooges Interference! Moe, Larry and Curly stand in a line with a spacing of 1.0 m. Larry is 3.0 m in front of a pair of stereo speakers 0.8 m apart, as shown below. The speakers produce a single-frequency tone, vibrating in phase with each other. What is the lowest possible frequency of the speakers that will allow all three of them to hear a loud sound (constructive interference)?

1 2 L1L1 L2L2 Larry is along the center line (m = 0), and will hear a loud spot no matter what. In order to solve this problem, we first need to use some trig to determine how far Curly and Moe are from each speaker.

1 2 L1L1 A lil’ trig 3 m 0.6 m L 1 = 3.06 m L 2 = 3.31 m

1 2 L1L1 L2L2 L 2 – L 1 = 0.25 m For constructive interference,

1 2 L1L1 L2L2 The lowest frequency will correspond with the highest wavelength. λ = 0.25 / m Which will correspond to m = 1, and λ = 0.25

λ = 0.25 m v = λf v sound = 343 m/s f = 1,372 Hz