Waves, Sound and Optics - wave behaviour, resonance and image formation
Waves, Sound and Optics
Wave questions test the wave equation, superposition (stationary waves and beats), the resonance-tube measurement of the speed of sound, and geometrical optics: refraction, total internal reflection, lenses and the prism. Expect numericals on lens position and magnification, critical angle, and the first resonance length of a tube.
Core rules
- v = fλ. Frequency is fixed by the source; speed depends on the medium. Transverse waves (light, waves on a string) and longitudinal waves (sound) behave differently under polarisation.
- Superposition: constructive when the path difference is nλ, destructive when it is (n + ½)λ. Beats frequency f_b = |f₁ − f₂|.
- Resonance tube: first resonance at l₁ ≈ λ/4, second at l₂ ≈ 3λ/4, so v = 2f(l₂ − l₁); with end correction, l + 0.3d = λ/4.
- Refraction: n = sin i / sin r = c/v; n₁ sin θ₁ = n₂ sin θ₂. Total internal reflection needs light travelling from the denser medium and i > C, where sin C = 1/n.
- Thin lens (real-is-positive convention): 1/f = 1/u + 1/v, magnification m = v/u = h_i/h_o; power P = 1/f in dioptres.
- Prism: at minimum deviation the ray passes symmetrically through the prism, and n = sin[(A + D)/2] / sin(A/2).
Worked example
An object is placed 15 cm from a convex lens of focal length 10 cm. Find the image and its magnification.
1/v = 1/f − 1/u = 1/10 − 1/15 = 1/30, so v = 30 cm. Since v is positive the image is real, formed on the other side of the lens; m = v/u = 30/15 = 2, so it is inverted and twice the object height.
Traps
Using degrees or centimetres inconsistently in v = fλ; writing n = sin r / sin i; ignoring the end correction in a resonance-tube result; forgetting that a converging lens can give a virtual image when u < f; claiming total internal reflection when light goes from air into glass; mixing the prism's refracting angle A with the deviation D.