Fluid Dynamics

Continuity conserves volume flow for incompressible fluids: $A_1v_1 = A_2v_2$ so narrowing a vessel speeds flow. Bernoulli's equation, $P + \tfrac{1}{2}\rho v^2 + \rho gh = \text{constant}$ is energy conservation for fluids: faster flow means lower pressure, the physics behind airflow, atherosclerotic vessels, and Venturi meters.

Waves and Sound

All waves obey $v = f\lambda$; speed is set by the medium, frequency by the source. Sound is longitudinal, with loudness measured logarithmically in decibels. The Doppler effect raises perceived frequency on approach and lowers it on recession (basis of Doppler ultrasound). Standing waves select discrete harmonics; a pipe closed at one end supports only odd harmonics.

Key Takeaways

• Depth, not container shape, sets static pressure.
• Continuity plus Bernoulli: narrower means faster means lower pressure.
• Buoyancy depends on displaced fluid, not the object's weight.
• Doppler shifts encode relative motion between source and observer.