Transmission & Reflection of Waves

Geometric Rule: For waves hitting a flat boundary, the angle of incidence ($heta_i$) is always equal to the angle of reflection ($heta_r$), both measured relative to the 'normal' (an imaginary line perpendicular to the surface).

Surface Texture: Smooth surfaces produce 'specular' reflection where waves reflect in a single predictable direction. Rough surfaces cause 'diffuse' reflection, scattering the wave energy in many directions.

Phase Changes: Depending on whether the second medium is 'denser' or 'lighter' than the first, the reflected wave may undergo a $180^{\circ}$ phase shift (inversion) or no phase shift at all.

Core Methodology: The pulse-echo technique involves emitting a short pulse of wave energy and measuring the time interval ($\Delta t$) it takes for the reflected 'echo' to return from a boundary.

Distance Calculation: The distance ($d$) to the boundary is calculated using the formula $d = \frac{v \cdot \Delta t}{2}$. The factor of 2 accounts for the wave traveling to the boundary and back.

Resolution: The ability to distinguish between two close boundaries depends on the wavelength; shorter wavelengths (higher frequencies) generally provide higher spatial resolution.

The 'Factor of 2' Rule: In pulse-echo problems, always check if the question asks for the 'total distance traveled' (which is $v \cdot t$) or the 'depth of the object' (which is $\frac{v \cdot t}{2}$).

Constant Frequency: Never change the frequency of a wave when it crosses a boundary. If the speed doubles, the wavelength must double.

Energy Conservation: Remember that the sum of reflected, transmitted, and absorbed energy must equal the incident energy. If a boundary is 'highly reflective,' the transmitted wave will have a very low amplitude.

Unit Consistency: Ensure wave speed is in meters per second ($m/s$) and time is in seconds ($s$) before calculating distances in meters ($m$).