7. Summary

Wave Eqn (String) ∂²y/∂t² = T/μ ∂²y/∂x²

The wave equation for a string relates the acceleration of the displacement of the string to the tension and mass per unit length. The equation states that the second derivative of displacement with respect to time is equal to the tension divided by the mass per unit length times the second derivative of displacement with respect to position. This shows how waves propagate through a medium (in this case, a string) and is fundamental in understanding wave mechanics.

Think of the wave equation like stretching a rubber band. When you pluck it, the way the energy travels through the band and returns is similar to how waves travel on a string, driven by the tension and mass of the rubber band.

Reflection/Transmission Depends on impedance mismatch

When a wave encounters a boundary between two different media, part of the wave will reflect back into the original medium, and part will transmit into the second medium. The amount of each depends on the impedance of the two media. Impedance is a measure of how much a medium resists the passage of a wave. If the impedances are mismatched, more of the wave reflects back, while less transmits forward.

Imagine throwing a ball against a wall. If the wall is concrete (high impedance compared to soft foam), the ball will bounce back (reflect), while if you throw it against a soft foam wall, it may absorb some of that energy and not bounce back much (transmit).

Standing Waves f_n = n v/(2L) for fixed strings

Standing waves occur when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other. For fixed strings, the formula f_n = n v/(2L) tells us that the frequencies of the standing waves are quantized. This means only certain frequencies can exist, which are determined by the length of the string and the wave speed.

Consider the strings on a guitar. When you pluck a string, it vibrates in a standing wave pattern, producing a specific note. Each note corresponds to a particular frequency, similar to our equation, showing that only certain notes can be played based on the string's length and the tension.

Longitudinal Waves Displacement along direction of travel

In longitudinal waves, the displacement of the particles in the medium is parallel to the direction of wave propagation. This is opposite to transverse waves, where displacement is perpendicular. This type of wave is particularly evident in sound waves moving through air.

Think of a slinky toy. If you push and pull one end of the slinky along its length, the coils compress and expand along the direction of your push. This motion illustrates how sound waves move through air, creating areas of compression and rarefaction.

Acoustics v = √(B/ρ)

In acoustics, the speed of sound in a medium is determined by the material's bulk modulus (B) and density (ρ). The formula describes how sound travels through different materials – faster in denser materials or those that are less compressible.

Imagine shouting in a swimming pool versus an empty field. The sound travels faster in the pool (more water and less air gaps) compared to the open air, reflecting how density and compressibility affect wave speed.

Dispersion v_p ≠ v_g; wave shape changes

Dispersion occurs when waves of different frequencies travel at different speeds. The phase velocity (v_p) refers to how fast a wave crest moves, while the group velocity (v_g) refers to the speed at which the overall shape of the wave packet moves. This difference leads to the spreading out of waves over time.

Think of a rainbow formed by light traveling through a prism. Each color bends at a different angle due to different speeds in the medium, much like how different sound frequencies can separate when they travel through the air.

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