16.6 Standing Waves and Resonance
3 min read•june 24, 2024
Standing waves form when identical waves traveling in opposite directions interfere. They create fixed nodes and antinodes, appearing stationary. This phenomenon occurs in various media, like guitar strings and pipe organs, producing distinct patterns of vibration.
amplifies oscillations when a system is driven at its . It's crucial in musical instruments for sound production and amplification. However, can also be dangerous in structures, potentially causing damage if not properly managed by engineers.
Standing Waves
Formation of standing waves
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- Two identical waves traveling in opposite directions interfere constructively and destructively at specific points
- Waves have the same , , and enabling
- creates antinodes with maximum displacement (crests and troughs align)
- creates nodes with no displacement (crests align with troughs)
- Standing waves appear stationary because nodes and antinodes remain fixed in position
- No net energy transfer occurs along the medium
- Distance between adjacent nodes or antinodes equals half the ()
- Standing waves form in various media
- Strings fixed at both ends (guitar strings, violin strings)
- Air columns open at one end and closed at the other (pipe organs) or closed at both ends (flutes)
- Membranes fixed along the edges (drumheads, speaker diaphragms)
Modes and nodes on strings
- Each mode of a on a string corresponds to a specific frequency and wavelength
- (1st ) has the lowest frequency with one at the center and nodes at the ends
- Wavelength equals twice the string length ()
- Frequency equals wave speed divided by twice the string length ()
- Higher harmonics (2nd, 3rd, etc.) have frequencies that are integer multiples of the
- Wavelengths equal twice the string length divided by the harmonic number ()
- Frequencies equal the harmonic number multiplied by the fundamental frequency ( or )
- (1st ) has the lowest frequency with one at the center and nodes at the ends
- Number of nodes equals the harmonic number plus one ()
- Fundamental mode has 2 nodes, 2nd harmonic has 3 nodes, etc.
- Number of antinodes equals the harmonic number ()
- Fundamental mode has 1 , 2nd harmonic has 2 antinodes, etc.
- represent the specific patterns of vibration that satisfy the of the system
Wave function and boundary conditions
- The describes the displacement of the medium at any point and time
- Boundary conditions determine how the wave behaves at the ends of the medium (e.g., fixed or free ends)
- The combination of the and boundary conditions defines the possible patterns
Resonance
Resonance in real-world applications
- Resonance amplifies oscillations when a system is driven at its natural frequency
- Energy efficiently transfers from the driving force to the system causing large vibrations
- Musical instruments rely on resonance to produce and amplify sound
- String instruments (guitars, violins) have natural frequencies determined by string length, tension, and mass per unit length
- Wind instruments (flutes, clarinets) have natural frequencies determined by air column length and end conditions (open or closed)
- Resonance boxes (guitar bodies, violin bodies) amplify sound by resonating at the same frequencies as the strings
- Bridges and structures have natural frequencies depending on material, size, and shape
- External forces (wind, earthquakes, marching soldiers) with frequencies matching natural frequencies can induce resonance
- Large-amplitude vibrations may cause structural damage or collapse (Tacoma Narrows Bridge, 1940)
- Engineers design structures to avoid resonance by ensuring natural frequencies differ from expected external frequencies
- External forces (wind, earthquakes, marching soldiers) with frequencies matching natural frequencies can induce resonance
- occurs when an external periodic force is applied to a system, potentially leading to resonance
- reduces the amplitude of oscillations over time, affecting the resonance behavior of a system
Key Terms to Review (43)
Amplitude: Amplitude is the maximum displacement of a point on a wave from its equilibrium position. It is a measure of the energy carried by the wave.
Amplitude: Amplitude is the maximum displacement or extent of a periodic motion, such as a wave or an oscillation, from its equilibrium position. It represents the magnitude or size of the motion and is a fundamental characteristic of various physical phenomena described in the topics of 1.7 Solving Problems in Physics, 8.4 Potential Energy Diagrams and Stability, 15.1 Simple Harmonic Motion, and beyond.
Antinode: An antinode is a point in a standing wave where the amplitude of oscillation is at its maximum. Antinodes occur at positions where constructive interference occurs.
Antinode: An antinode is a point in a standing wave pattern where the amplitude of the wave is at its maximum. It is the opposite of a node, where the amplitude is at its minimum.
Boundary Conditions: Boundary conditions are the set of constraints or specifications that define the physical environment or system at the boundaries of a problem. They are crucial in determining the behavior and characteristics of various physical phenomena, such as standing waves and normal modes of sound waves.
Chladni: Chladni is a German physicist who is known for his pioneering work in the field of acoustics, particularly in the study of standing waves and resonance. His experiments with vibrating plates and the patterns they produce have become an important tool in the understanding of wave phenomena.
Constructive interference: Constructive interference occurs when two or more waves superimpose to form a resultant wave with a greater amplitude than any of the individual waves. This happens when the phase difference between the waves is an integer multiple of $2\pi$ radians.
Constructive Interference: Constructive interference occurs when two or more waves combine in such a way that their amplitudes add together, resulting in a larger amplitude at the point of intersection. This phenomenon is observed in various wave-based phenomena, including those related to energy and power of waves, interference of waves, standing waves and resonance, normal modes of standing sound waves, and beats.
Damping: Damping refers to the process of reducing or controlling the amplitude or oscillation of a system over time. It is a phenomenon that occurs in various physical systems, including mechanical, electrical, and electronic systems, where it serves to dissipate energy and prevent excessive vibrations or oscillations.
Destructive interference: Destructive interference occurs when two waves superimpose to form a resultant wave of lower amplitude. This happens when the crest of one wave aligns with the trough of another, effectively canceling each other out.
Destructive Interference: Destructive interference occurs when two waves interact in such a way that their amplitudes cancel each other out, resulting in a decrease or complete elimination of the wave's intensity at certain points in space. This phenomenon is a fundamental principle in the study of wave behavior and has important applications across various fields, including physics, engineering, and acoustics.
Diffraction: Diffraction is the bending or spreading of waves around the edges of an obstacle or through an aperture. It is a fundamental phenomenon in the behavior of waves, including light, sound, and matter waves, and is closely related to the concepts of interference and standing waves.
Forced Oscillation: Forced oscillation refers to the phenomenon where an external force or driving force causes a system to oscillate at a specific frequency, even if the system's natural frequency is different. This concept is crucial in understanding both simple harmonic motion and the behavior of standing waves.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Fundamental frequency: The fundamental frequency is the lowest frequency at which a system oscillates. It is also known as the first harmonic and determines the pitch of a sound in musical instruments.
Fundamental Mode: The fundamental mode, also known as the first mode or the ground state, refers to the lowest-energy standing wave pattern that can be established in a system. This mode is the simplest and most basic configuration of a standing wave, and it is the first mode that is typically observed in various physical systems, including those involving standing waves and resonance.
Harmonic: A harmonic is a frequency that is an integer multiple of the fundamental frequency in a periodic waveform. Harmonics are essential in the study of standing waves and resonance, as they determine the patterns and frequencies of vibration in various physical systems.
Interference: Interference is the phenomenon that occurs when two or more waves, such as sound or light waves, interact with each other. This interaction can result in the reinforcement or cancellation of the waves, depending on the relative phases of the waves.
Kundt's Tube: Kundt's tube is a device used to study the properties of standing waves and to determine the speed of sound in a gas. It consists of a long, closed-ended tube filled with a gas, typically air, and a mechanism to generate sound waves at one end of the tube.
Linear wave equation: The linear wave equation is a second-order partial differential equation that describes the propagation of linear waves, such as sound or light waves, in a medium. It is typically written as $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$, where $u$ represents the wave function and $c$ is the speed of the wave.
Longitudinal wave: A longitudinal wave is a type of wave where the displacement of the medium is parallel to the direction of wave propagation. Sound waves in air are common examples of longitudinal waves.
Longitudinal Wave: A longitudinal wave is a type of wave in which the oscillation of the medium is parallel to the direction of wave propagation. In other words, the particles of the medium move back and forth in the same direction as the wave is traveling.
Natural Frequency: Natural frequency is the inherent frequency at which a system tends to oscillate when it is not affected by external forces. It is a fundamental property of a system that depends on its physical characteristics and determines how the system will respond to various inputs or disturbances.
Node: A node is a point along a standing wave where the wave has minimal or zero amplitude due to destructive interference. Nodes occur at fixed intervals and are points of no displacement.
Node: A node is a point along a standing wave where the wave's amplitude is zero, indicating the presence of destructive interference. Nodes are crucial in the understanding of standing waves and resonance phenomena.
Normal mode: A normal mode is a pattern of motion in which all parts of a system oscillate with the same frequency and in a fixed phase relation. In mechanical systems, these modes are often standing waves.
Normal Modes: Normal modes are specific patterns of vibration that occur in a system when it oscillates at particular frequencies. These modes are characterized by the fact that all parts of the system vibrate with the same frequency, but with different amplitudes and phases, creating a fixed shape that doesn't change over time. Understanding normal modes is crucial for analyzing standing waves and resonance phenomena, particularly in sound waves where distinct frequencies result in different pitches.
Overtone: An overtone is a higher frequency standing wave mode that occurs in addition to the fundamental frequency. Overtones contribute to the harmonic series and influence the timbre of sound.
Principle of superposition.: The principle of superposition states that when two or more waves overlap in space, the resultant displacement is the algebraic sum of the individual displacements. This principle applies to all linear systems.
Resonance: Resonance occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude. It is a crucial concept in oscillations and wave phenomena.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, leading to a significant increase in the amplitude of the system's response. This concept is fundamental across various fields in physics, including mechanics, acoustics, and electromagnetism.
Snell's law: Snell's law describes how light refracts when it passes from one medium to another, defining the relationship between the angles of incidence and refraction based on the indices of refraction of the two media. This principle is crucial in understanding how waves behave at boundaries, particularly when considering wave interference patterns and resonance in various contexts. It allows us to predict how light bends, which is essential for applications like lenses and optical devices.
Standing wave: A standing wave is a wave that remains in a constant position and does not appear to travel through the medium. It is formed by the interference of two waves traveling in opposite directions with the same frequency and amplitude.
Standing Wave: A standing wave is a wave pattern that occurs when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other, resulting in a stationary wave pattern with fixed points of constructive and destructive interference.
Superposition: Superposition is the principle that describes how waves, such as sound or light, can combine to form a new wave. It states that when two or more waves interact, the resulting wave is the sum of the individual waves, both in terms of amplitude and phase.
Transverse wave: A transverse wave is a type of wave where the oscillations or vibrations are perpendicular to the direction of the wave's advance. Examples include waves on a string and electromagnetic waves.
Transverse Wave: A transverse wave is a type of wave where the oscillation of the medium is perpendicular to the direction of wave propagation. This means that the particles in the medium move back and forth in a direction that is at right angles to the way the wave is traveling.
Tubes with anti-symmetrical boundary conditions: Tubes with anti-symmetrical boundary conditions are tubes in which one end is closed and the other end is open. These conditions create distinctive standing wave patterns and specific harmonic frequencies.
Wave Equation: The wave equation is a fundamental mathematical equation that describes the propagation of waves, such as sound waves, light waves, and waves on a string. It governs the relationship between the displacement of a wave and the variables that determine its behavior, including time, position, and the properties of the medium through which the wave is traveling.
Wave function: A wave function is a mathematical representation that describes the amplitude of a wave at each point in space and time. It encapsulates the properties of waves such as frequency, wavelength, and phase.
Wave Function: The wave function is a mathematical representation of the quantum state of an object or a system. It provides a complete description of the quantum mechanical behavior of an entity and allows for the calculation of the probability of its measurable properties.
Wavelength: Wavelength is the distance between successive crests or troughs of a wave. It is typically represented by the Greek letter lambda ($\lambda$).
Wavelength: Wavelength is a fundamental characteristic of waves, representing the distance between consecutive peaks or troughs of a wave. It is a crucial parameter that describes the spatial properties of various wave phenomena, including light, sound, and other types of oscillations.