Final answer:
To rank the waves by phase velocity, we calculate the velocity for each wave using available information, such as wavelength, frequency, angular frequency, and wave number. The waves are then ordered from the lowest to the highest phase velocity: Waves 2, 1, 3, 5, and 4, with wave 6 being unranked due to insufficient information.
Step-by-step explanation:
To order the waves by their phase velocity, we need to determine the speed of each wave based on the given equations. The phase velocity (v) of a wave can be calculated with the formula v = λf, where λ is the wavelength and f is the frequency of the wave. Alternatively, if the angular frequency (ω) and the wave number (k) are given, the phase velocity can also be found using v = ω/k.
d(x, t) = (2.0 m) sin([25 / m] x - [1.2 / s]t) gives ω = 1.2 s⁻¹ and k = 25 m⁻¹, so v = 1.2 / 25 = 0.048 m/s.
The second wave has ω = 0.0012 rad/s and λ = 36.75 m, so v = 0.0012 * 36.75 = 0.0441 m/s.
The third wave has a wavelength of 3.75 m and a period of 4.90 s, so f = 1 / T = 1 / 4.90 s and v = f λ = (1 / 4.90) * 3.75 = 0.7653 m/s.
d(x, t) = (5.6 mm) sin([(2 * π) / (3.0 m)]x - [(2 * π) / (0.00036 s)]t) gives v = (2 * π) / (0.00036 * 3.0) = 17477.78 m/s.
d(x, t) = (0.05 m) cos([0.48 / m]x + [5.2 / s]t) gives v = 5.2 / 0.48 = 10.833 m/s.
For the sixth wave, we cannot calculate the velocity directly from the given equation, as it is a product of sine and cosine, hence not in a standard sinusoidal wave form.
Organizing the waves by phase velocity gives us the following order:
Wave 2 with v = 0.0441 m/s
Wave 1 with v = 0.048 m/s
Wave 3 with v = 0.7653 m/s
Wave 5 with v = 10.833 m/s
Wave 4 with v = 17477.78 m/s
Wave 6 - velocity unknown based on the given equation