Waves Test. 3

The beat frequency is f_beat = |f₁ - f₂| = |405 - 400| = 5 Hz. The time period (interval) between beats is the reciprocal of the beat frequency, so T = 1/f_beat = 1/5 = 0.2 s.

Waves are a mechanism for transferring energy without a net transfer of the medium itself. The particles of the medium oscillate but return to their equilibrium positions.

For a closed pipe, the harmonics are the odd integers (1st, 3rd, 5th, ...). The fundamental is the 1st harmonic. The first overtone is the 3rd harmonic. The second overtone is the 5th harmonic.

The energy from a point source spreads out over the surface of a sphere, whose area is 4πr². Since the total power is constant, the intensity (Power/Area) is inversely proportional to the square of the distance.

Particles at an antinode reach maximum displacement when particles at a node have zero displacement. Their velocity and displacement are 90 degrees out of phase.

For a string fixed at both ends, the number of segments (or antinodes) directly corresponds to the harmonic number. So, 5 segments is the 5th harmonic.

For a string, the fundamental is the 1st harmonic. The next frequency is the 2nd harmonic (first overtone). The one after that is the 3rd harmonic (second overtone).

Sound reaching the stationary cliff is Doppler-shifted because the source moves toward it: f_wall = f * v/(v - u) = 1000 * 330/(330 - 10) = 1031.25 Hz. The cliff (stationary) reflects this frequency unchanged, and the stationary observer receives it with no further shift. Hence the echo ≈ 1031 Hz

Since the speed of sound v is proportional to the square root of the absolute temperature T (v ∝ √T), quadrupling T (making it 4T) will change the speed to √4v = 2v.

As the particle passes through its equilibrium position, the restoring force is zero, but its kinetic energy and speed are at their maximum.

When the driving frequency matches the object's natural frequency of vibration, energy is transferred most efficiently, leading to a large increase in the amplitude of oscillation. This is resonance.

In a stationary wave, energy is not propagated. Instead, it oscillates between kinetic energy of the particles and potential energy of the medium, localized within each loop (between nodes).

When reflecting from a boundary that is free to move (a 'soft' reflection), the wave does not get inverted. It reflects with no change in its phase.

For an open pipe, all harmonics are present. The fundamental is f₁ = v/2L = 340/1 = 340 Hz. The third harmonic is f₃ = 3f₁ = 3 × 340 = 1020 Hz.

The Wave number (propagation constant), k, is related to the wavelength by k = 2π/λ. It describes the spatial variation of the wave.

A full wavelength (λ) corresponds to a full cycle, which is a phase of 2π radians. Therefore, a quarter wavelength (λ/4) corresponds to a quarter of a cycle, which is a phase difference of 2π/4 = π/2 radians (90°).

Distant galaxies are moving away from us. This relative motion causes the light waves to be 'stretched', shifting their frequency towards the red end of the spectrum (lower frequency, longer wavelength).

The first resonance occurs at approximately L₁ = λ/4 and the second at L₂ = 3λ/4. The difference between them is L₂ - L₁ = 3λ/4 - λ/4 = λ/2. Therefore, λ = 2(L₂ - L₁).

The amplitude is defined as the maximum displacement or distance moved by a point on the wave from its equilibrium position.

Using the wave equation v = fλ, we can rearrange to find the wavelength: λ = v/f = 340 / 500 = 0.68 m.