Resonance

Travelling Waves

Pulse on a string

Can have a wave travelling down the string OR can have a standing wave

Transverse Oscillations

String moves up and down forming a wave that travels along the string.

Similar to waves in the ocean.

Longitudinal Oscillation

Sound is an example of this type of wave. Region of low pressure and high pressure

• Pressure is a force exerted over an area
• Pressure = Force divided by Area
• Strong force on a large area has same pressure as weak force on a small area

Speed of Wave

Speed (\(v\))

Frequency \((f)\): The number of oscillations that occur in a second

Period \((T)\): The time it takes for one oscillation to occur

Speed of travelling wave = Frequency \(\times\) Wavelength

• \(v = L \times f\)

Why does string move up and down?

• Restoring force from tension in the string

Doppler Effect: Change in Train Whistle

• There is an error in the slide. The middle train is moving right, TOWARDS you (not away).
• Speed of sound constant
• Waves get compressed or expanded

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1. Train Not Moving

• Wavelength \((L)\): The distance between two consecutive crests or troughs in a wave is consistent.
• Frequency \((f)\): The number of wave crests passing a fixed point per unit time remains unchanged.
• This represents the baseline condition where the sound heard by an observer is the same as the sound produced because there is no relative motion between the source and the observer.

2. Train Moving Right, Towards You

• Wavelength Decreases: As the train moves away from the observer, each successive wave crest is emitted from a position further to the right than the previous one, reducing the distance between crests in the direction of the observer.
• Frequency Increases: The compression of the wave crests means they pass the observer more frequently, leading to a higher perceived frequency. This is why a sound, like a train whistle, seems higher-pitched as it starts to move away but is still approaching you.

3. Train Moving Left, Away from You

• Wavelength Increases: Here, the train is moving away but to the left of the observer. Each wave crest is emitted further to the left, increasing the distance between crests in the direction of the observer.
• Frequency Decreases: The expansion of the wave crests means they pass the observer less frequently, leading to a lower perceived frequency. This results in the sound, like the train whistle, appearing lower-pitched as it moves away.

Key Concept: Speed of Sound Remains Constant

• Despite the train's movement altering the wavelength and frequency of the sound waves, the speed of sound itself remains constant in the medium (air, in this case). The changes in frequency and wavelength are due to the relative motion of the source and the observer, not due to any change in the intrinsic speed at which sound waves propagate.

The Doppler Effect is a critical concept in physics and is used in various applications, including radar and medical imaging, and it also helps explain everyday observations like the changing pitch of sirens or train whistles as they move towards and then away from an observer.

Pendulum

• String pulls the mass to the right
  • The mass keeps moving, past the bottom and off to the right.
  • The string then pulls the mass to the left.
  • The mass goes back and forth.
• Restoring Force:
  • Always tries to bring the mass to equilibrium.
• Simple Harmonic Oscillator:
  • Characterized by the mass on a string in periodic motion.
• Period of Pendulum:
  • Time to go back and forth depends only on the length of the string.
  • Long string results in a long period and low frequency.
    • (This means the pendulum swings more slowly with a longer string, corresponding to a lower frequency of oscillation.)

Harmonic Oscillator

• Start with spring stretched
  • Spring pulls mass to the left
  • Spring gets squeezed and pushes the mass to the right
  • Spring applies a restoring force to the mass (tries to place the mass so the string is unstretched and relaxed)
• Frequency of Oscillation:
  • Depends on the mass and the stiffness of the spring
  • Bigger mass results in lower frequency
  • Stiffer spring results in higher frequency

Resonance

Example: Pushing a child on a swing

• Natural Frequency: Oscillators have a natural frequency at which they prefer to oscillate.
• Resonant Frequency: This is the resonant frequency of the oscillator.
• Energy Application: If energy is applied to the oscillator at the same frequency, the amplitude of the oscillation will increase.

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Natural Frequency and Resonant Frequency

• Natural Frequency: This is the frequency at which a system naturally oscillates when it is disturbed and then allowed to move freely without further external forces acting on it. Every mechanical system has a natural frequency determined by its physical properties.
• Resonant Frequency: This is essentially the same as the natural frequency for a simple system. It's the frequency at which the system, if subjected to periodic driving forces, will exhibit the maximum amplitude of oscillation.

How Resonance Works

• The image shows one person pushing another on a swing, which is a familiar and relatable example of resonance:
  • If you push the swing exactly in sync with its natural swinging motion (i.e., at its natural or resonant frequency), each push adds more energy to the swing's motion.
  • This synchronization maximizes the efficiency of energy transfer between the pusher and the swing, causing the amplitude (the height of the swing's arc) to increase significantly with each push.

Energy Application at Resonant Frequency

• If energy is applied at the same frequency: This refers to external forces (in this case, the pushes) being applied at intervals that match the swing's natural oscillation period.
• Amplitude Increase: Because the energy is added at the moment when it can most effectively enhance the swing's motion (in phase with it), the amplitude of the swing's oscillations increases. This is due to constructive interference, where the push reinforces the swing's motion.

Standing Waves

String has two fixed ends

Standing Waves on a string fastened at both ends (Violin)

• Stationary Ends: The string at the ends cannot move up or down (remains stationary).
• Wave Properties:
  • Speed of the wave along the string is constant.
  • Different wavelengths result in different frequencies, known as harmonics.

Sounds from Wine Glass

• Friction on Glass Rim: Rubbing a finger along the rim of a glass generates friction.
• Stick/Slip Phenomenon: This frictional interaction causes the rim of the wine glass to move.
• Influence of Glass Properties: The size and shape of the glass determine its natural frequency.
• Energy Addition: Rubbing the finger adds energy, amplifying the sound, similar to using a bow on a violin.

• Bowl Movement: The bowl of the wine glass moves in and out like a water balloon, pushing air back and forth and producing a sound wave (pressure wave).

  

Sounds from a Tube (Organ Pipe)

• Pressure Differences: Pressure differences in the tube provide a restoring force.
• Simple Harmonic Oscillator: The system behaves like a simple harmonic oscillator.
• Fundamental Wavelength: The fundamental wavelength is twice the length of the tube.
• Air Movement: Air moves in and out at the ends of the tube.
• Energy Input: Blowing air across the bottom of the tube adds energy to the resonator, though the coupling process is complicated.

Sounds from a beer bottle

• Comparison: Behaves like a mass on a spring.
  • Air in the bottle acts like the mass.
  • Air in the large part of the bottle provides the restoring forces as it expands and contracts.
• Analogy: This is similar to a spring system where the mass moves back and forth due to the spring's restoring force.
• Effect of Liquid: Adding liquid to the bottle decreases the size of the air compartment, which changes the restoring force and therefore the frequency of the oscillations.

Loud Speaker

• The pressure wave is created by the speaker cone moving back and forth.

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Breaking a glass with sound using resonance

• Sound Waves and Resonance: Every object has a natural frequency, which is the frequency at which it naturally vibrates. For a wine glass, this natural frequency depends on factors like its shape, size, and material. When sound waves at this specific frequency are generated near the glass, they cause it to vibrate at its natural frequency.
• Speaker Producing Sound Waves: The image shows a speaker, likely driven by an electromagnet, producing sound waves. The speaker cone moves back and forth, creating sound waves in the air. If the frequency of these sound waves matches the natural frequency of the glass, resonance occurs.
• Wine Glass Oscillation: Resonance amplifies the oscillations in the glass. The sound waves cause the glass to oscillate in and out, similar to how a balloon expands and contracts. With each cycle, the glass vibrates more intensely because the energy from each wave adds to the previous vibration.
• Increasing Amplitude and Breaking Point: As the resonance builds up, the amplitude of the vibrations in the glass increases. If the sound continues at the resonant frequency, the vibrations can grow strong enough to exceed the structural limits of the glass. When this happens, the glass can no longer withstand the stress and eventually shatters.