What are the wavelength limits in water for light with wavelengths from 400 nm to 700 nm in air, and what is the speed of a transverse wave on a 64 cm steel wire of mass 5 g under 8 N tension?

Wavelength range in air: $\lambda_{air, \text{min}} = 400 \text{ nm}$, $\lambda_{air, \text{max}} = 700 \text{ nm}$.
Refractive index of water: $n = 1.33$.

The formula is $n = \frac{\lambda_{air}}{\lambda_{water}}$, which means $\lambda_{water} = \frac{\lambda_{air}}{n}$.

For the minimum wavelength:

For the maximum wavelength:

The limits of the wavelength in water are:

Part b(ii): Speed of a transverse wave on a steel wire

Length of wire: $L = 64 \text{ cm} = 0.64 \text{ m}$.
Mass of wire: $m = 5 \text{ g} = 0.005 \text{ kg}$.
Tension force: $F = 8 \text{ N}$.

The formula for the speed of a transverse wave on a string is $v = \sqrt{\frac{F}{\mu}}$.

The speed of the transverse wave is:

Part 18a(i): Definitions of pitch, resonance, and Doppler effect

• Pitch: The characteristic of a sound that determines whether it is high or low, primarily determined by the frequency of the sound wave. Higher frequency corresponds to higher pitch.

• Resonance: A phenomenon that occurs when an oscillating system is driven at its natural frequency, resulting in a large amplitude of oscillation. This happens when the frequency of the applied force matches the natural frequency of the system.

• Doppler effect: The apparent change in frequency or wavelength of a wave (sound or light) observed by an observer moving relative to its source. The frequency appears higher when the source and observer are approaching each other, and lower when they are moving apart.

Part 18a(ii): Depth of the sea and wavelength of ultrasound

Frequency of ultrasound: $f = 50 \text{ kHz} = 50 \times 10^3 \text{ Hz}$.
Time for round trip: $t = 0.80 \text{ s}$.
Speed of sound in sea water: $v = 1500 \text{ m/s}$.

The signal travels to the ocean bed and back, so the distance covered is $2 \times \text{depth}$.
Distance = speed $\times$ time.

The formula relating speed, frequency, and wavelength is $v = f\lambda$.

The depth of the sea is $\boxed{600 \text{ m}}$ and the wavelength of the signal is $\boxed{0.03 \text{ m}}$.

Part b(i): Frequency heard by the observer on the other train

Speed of trains (source and observer): $v_s = v_o = 90 \text{ kmh}^{-1}$.
Convert to m/s: $90 \text{ kmh}^{-1} = 90 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 25 \text{ m/s}$.
Source frequency: $f_s = 520 \text{ Hz}$.
Speed of sound in air: $v = 350 \text{ m/s}$.

When the source is approaching the observer, the denominator is $(v - v_s)$.
When the observer is approaching the source, the numerator is $(v + v_o)$.
The formula for the observed frequency $f_o$ is:

The frequency heard by the observer on the other train is:

Part b(ii): Atmospheric temperature

Speed of sound at temperature $T$: $v_T = 336 \text{ m/s}$.
Speed of sound at $0^\circ\text{C}$: $v_0 = 332 \text{ m/s}$.

The speed of sound in air increases by approximately $0.6 \text{ m/s}$ for every $1^\circ\text{C}$ rise in temperature above $0^\circ\text{C}$.
The formula is $v_T = v_0 + 0.6T$, where $T$ is the temperature in $^\circ\text{C}$.

The atmospheric temperature is: