Interactive physics simulator
Longitudinal Wave
Observe the dynamics of compression waves. Study compressions and rarefactions in a gas tube, calculate the speed of sound across different gases and temperatures, and map the 90° phase shift between particle displacement and pressure waves.
Longitudinal Wave Laboratory
Interact with the sliders to change parameters in real-time. Check 'Freeze Frame' to measure with calipers.
Live Wave Telemetry
- Wave Speed (v)
- 343.0 m/s
- Wavelength (λ)
- 0.34 m
- Frequency (f)
- 1000 Hz
- Displacement (s)
- 0.00 cm
- Caliper Distance
- 0.00 m
Introduction to Longitudinal Waves
A longitudinal wave is a wave in which the particles of the medium oscillate or vibrate parallel to the direction of the wave\'s propagation. While a transverse wave oscillates perpendicular to its travel, a longitudinal wave moves in the same plane as the energy transfer.
The most common example of a longitudinal wave is a sound wave traveling through air. As the sound source (like a speaker cone) vibrates back and forth, it compresses and expands the adjacent air molecules. These molecules collide with their neighbors, transmitting the pressure disturbance through the air while the molecules themselves only shake back and forth locally.
The Anatomy of a Longitudinal Wave
Instead of crests and troughs, a longitudinal wave is characterized by alternating regions of particle density and pressure:
- Compression: A region where the particles of the medium are squeezed closest together. Compressions represent areas of high density and maximum positive pressure variation (+ΔP).
- Rarefaction: A region where the particles are spread furthest apart. Rarefactions represent areas of low density and maximum negative pressure variation (-ΔP).
- Wavelength (λ): The spatial distance between the centers of two consecutive compressions, or two consecutive rarefactions.
- Displacement Amplitude (smax): The maximum distance an individual medium particle moves from its equilibrium position.
- Pressure Amplitude (Δpmax): The maximum change in pressure from the medium\'s normal atmospheric pressure.
Speed of Sound in Gases
The speed of sound v is determined by the physical characteristics of the gas medium. In an ideal gas, the speed at which a compression signal travels depends on the adiabatic index γ, the universal gas constant R, the absolute temperature T (in Kelvin), and the molar mass M (in kg/mol):
Where:
- γ (Gamma): The adiabatic index (approx. 1.40 for diatomic gases like Air, and 1.67 for monatomic gases like Helium).
- R: The universal gas constant, equal to 8.314 J/(mol·K).
- T: The absolute temperature in Kelvin (TK = T°C + 273.15). Higher temperature increases molecular speeds, increasing sound velocity.
- M: The molar mass of the gas. Lighter gases (like Helium, M = 4 g/mol) allow sound to propagate much faster than in air (M = 29 g/mol) at the same temperature.
The Phase Relationship: Displacement vs. Pressure
A key concept in acoustic physics is that the displacement wave s(x, t) and the pressure wave P(x, t) are exactly 90° out of phase (π/2 radians).
When gas molecules on both sides of a point move toward that point, the particles at the center do not move at all (displacement is zero). However, the accumulation of molecules at that point creates a maximum compression (maximum pressure).
Conversely, when particles are at their maximum displacement, they are moving in unison with no relative compression occurring. Thus:
- Displacement Nodes (s = 0) correspond to Pressure Antinodes (ΔP = ±ΔPmax).
- Displacement Antinodes (s = ±smax) correspond to Pressure Nodes (ΔP = 0).
This mathematical phase relationship explains how standing waves behave in wind instruments and sound resonators.
Solved Examples
A sound wave propagates through air at 20°C where its speed is 343 m/s. If the wavelength is measured to be 0.1715 meters, calculate the frequency of this sound wave. Is it within the audible range of human hearing?
View Step-by-Step Solution
- Identify the given values: speed v = 343 m/s, and wavelength λ = 0.1715 meters.
- Use the wave speed equation: v = f · λ.
- Solve for frequency f: f = v / λ.
- Substitute values: f = 343 m/s / 0.1715 m = 2000 Hz (or 2 kHz).
- The range of human hearing is roughly 20 Hz to 20,000 Hz.
- Since 2000 Hz is between 20 Hz and 20 kHz, it is within the audible range.
- The frequency is 2000 Hz and it is audible.
**Final Answer:** f = 2000 Hz (audible)
Calculate the speed of a longitudinal wave (sound) in water. The Bulk Modulus of water is 2.2 × 10&sup9; N/m² (or Pa), and its density is 1000 kg/m³.
View Step-by-Step Solution
- Identify the variables: Bulk Modulus B = 2.2 × 10&sup9; Pa, and density ρ = 1000 kg/m³.
- Use the speed of a longitudinal wave in a fluid formula: v = √(B / ρ).
- Substitute values: v = √(2.2 × 10&sup9; / 1000) = √(2.2 × 10&sup6;).
- Calculate: v ≈ 1483.2 m/s.
- The speed of the longitudinal wave in water is approximately 1483.2 m/s.
**Final Answer:** v ≈ 1483.2 m/s
Helium gas has a molar mass of M = 0.004 kg/mol and an adiabatic index of γ = 1.67. (a) Calculate the speed of sound in helium at 0°C (273 K). (b) Explain why sound travels faster in helium than in air at the same temperature (M_air ≈ 0.029 kg/mol, γ = 1.40). (Universal Gas Constant R = 8.314 J/(mol·K)).
View Step-by-Step Solution
- Identify variables for helium: M = 0.004 kg/mol, γ = 1.67, T = 273 K.
- Part (a): Use the speed of sound in an ideal gas formula: v = √(γ R T / M).
- Substitute values: v = √(1.67 · 8.314 · 273 / 0.004) ≈ √(3790.66 / 0.004) = √(947665) ≈ 973.5 m/s.
- Part (b): Even though the adiabatic index γ for air is slightly lower (1.40 vs 1.67), the molar mass of helium is much smaller (0.004 kg/mol vs 0.029 kg/mol).
- Because speed of sound is inversely proportional to the square root of molar mass (v ∝ √(1/M)), the lighter helium molecules travel much faster at a given temperature, transferring kinetic energy quicker.
- The speed of sound in helium at 0°C is ≈ 973.5 m/s, which is much faster than in air (≈ 331 m/s) due to helium's low molar mass.
**Final Answer:** vHe ≈ 973.5 m/s (faster than air due to lower molar mass)
Common Misconceptions & Pitfalls
- Misconception: Air molecules travel all the way from the speaker/speaker cone to your ear.
**Reality:** No. Air molecules only oscillate back and forth about their own local equilibrium positions by a tiny fraction of a millimeter. It is the collision energy (pressure wave) that travels from the speaker to your ear. - Misconception: Sound waves can travel in space since electromagnetic waves do.
**Reality:** No. Light waves are electromagnetic and don\'t require matter. Sound waves are mechanical longitudinal waves that require physical particle collisions to propagate. Outer space is a vacuum, so sound cannot travel. - Misconception: Displacement and pressure peak at the same locations in a sound wave.
**Reality:** They are 90° out of phase. A compression (max pressure) occurs at a displacement node where particles are rushing in from both sides, meaning the displacement itself at that exact point is zero.
Practice Questions
Question 1
Why does sound travel faster in solids than in liquids, and faster in liquids than in gases, despite solids having much higher densities?
Show Explanation
The speed of a longitudinal wave depends on the ratio of elastic properties to inertial properties: v = √(E / ρ) for solids, or v = √(B / ρ) for fluids. While solids have a higher density (ρ), which would tend to decrease the speed, their elastic stiffness (Young's Modulus E) is several orders of magnitude larger than that of liquids or gases. This high stiffness means adjacent atoms exert huge restoring forces on each other when squeezed, propagating the disturbance extremely rapidly. The increase in elastic stiffness far outweighs the increase in density, resulting in much higher speeds.
Question 2
Explain why a displacement node in a longitudinal wave corresponds to a pressure antinode, and vice versa.
Show Explanation
A displacement node is a point where medium particles do not move. On either side of this node, particles are moving in opposite directions. If they move toward the node, they compress the medium, creating a pressure peak (antinode). If they move away from the node, they create a rarefaction (pressure minimum). Therefore, a displacement node corresponds to maximum pressure variation. Conversely, a displacement antinode is a point where particles undergo maximum displacement, meaning they move in unison, keeping the local density and pressure constant (pressure node, zero variation). This results in a 90° (π/2 rad) phase shift.
Question 3
How does gas temperature affect the speed of sound, and why does wind not change this speed?
Show Explanation
In an ideal gas, the speed of sound is given by v = √(γ R T / M). As temperature (T) increases, the average kinetic energy of the gas molecules increases, meaning they collide more frequently and transfer the compression signal faster, increasing wave speed. Wind is the bulk movement of the entire air medium relative to the ground. Wind does not change the speed of sound relative to the air itself; it simply adds vectorially to the speed relative to a stationary ground observer.
Frequently Asked Questions
- What is a longitudinal wave?
- A longitudinal wave is a wave where particles of the medium oscillate parallel to the direction of the wave's travel.
- What are compressions and rarefactions?
- Compressions are dense areas of high pressure where particles are squeezed together. Rarefactions are sparse areas of low pressure where particles are spread apart.
- Why can longitudinal waves travel in fluids but mechanical transverse waves cannot?
- Longitudinal waves propagate through compression and expansion, which depend on the Bulk Modulus (resistance to volume changes). Fluids can resist compression. Transverse waves propagate through shear forces, but fluids cannot support shear stress (they flow instead of restoring shape), meaning transverse waves cannot propagate through them.
- What is the phase difference between pressure and displacement in sound?
- The pressure wave and displacement wave are exactly 90° (π/2 radians) out of phase. The points of maximum pressure variation (compressions/rarefactions) correspond to zero displacement.
- Can you hear sound in space?
- No. Sound is a mechanical longitudinal wave that requires a material medium (like gas, liquid, or solid) to propagate. Space is a vacuum with no particles to transfer collisions, so sound cannot travel there.