A surfing physicist paddles out beyond the breaking surf to a deep-water region where the ocean waves are sinusoidal (can obey a sine or cosine function) in shape, with crest 14 m apart. The surfer rises a vertical distance 3.6 m from wave trough to crest, a process that takes 1.5s. Find the wave speed, and describe it with an equation. Take a wave crest to be at x=0 at time = 0, with the positive x direction toward the open ocean.

The amplitude is A = 1.8 m, and the wavelength is L = 14 m. The period is P = 3.0 s. Use a cosine function to make the deflection a crest at t = 0 and x = 0.

y = A cos [2 pi (t/P - x/L]

Well, let's dive into the problem, shall we? The first thing we need to do is find the wave speed. To find the wave speed, we can use the equation:

wave speed = wavelength / period

where the wavelength is the distance between two consecutive crests and the period is the time it takes for one complete wave to pass a given point.

Given that the crest-to-crest distance is 14 m and the time it takes for one complete wave is 1.5 s, we can plug in the values:

wave speed = 14 m / 1.5 s

Now, let me put on my mathematical clown wig and pull out my calculator... *honk honk honk* crunching some numbers... and voila! The wave speed is approximately 9.33 m/s.

Now, let's describe this wave with an equation. We can use the equation of a sinusoidal wave:

y(x, t) = A * sin(kx - ωt)

Where:
- y represents the vertical displacement at position x and time t
- A is the amplitude, which in this case is half of the vertical distance from trough to crest (3.6 m / 2 = 1.8 m)
- k is the wave number, which is equal to 2π divided by the wavelength (2π / 14 m)
- x represents the horizontal position along the wave
- ω is the angular frequency, which is equal to 2π divided by the period (2π / 1.5 s)
- t represents the time

Putting it all together, the equation for this wave can be written as:

y(x, t) = 1.8 * sin((2π / 14) * x - (2π / 1.5) * t)

There you have it, a wave equation suitable for any physics-themed clown act. Enjoy, and ride those waves like a dancing clown on a unicycle!

To find the wave speed, we can use the equation:

wave speed (v) = wavelength (λ) / period (T)

Given that the wave crest is 14 m apart, this means the wavelength (λ) is 14 m.

The period (T) can be found by the time it takes for the surfer to rise from wave trough to crest, which is given as 1.5 s.

So, the period (T) is 1.5 s.

Using the equation for wave speed:

v = λ / T

Substituting the known values:

v = 14 m / 1.5 s

Calculating:

v ≈ 9.33 m/s

Therefore, the wave speed is approximately 9.33 m/s.

To describe the wave with an equation, we can use a sinusoidal function. Since the wave in this scenario can obey a sine or cosine function, we can choose either.

Using a sine function, the equation for the wave can be written as:

y(x, t) = A * sin(kx - ωt)

In the given scenario, the surfer rises a vertical distance of 3.6 m, which is the amplitude (A) of the wave.

The wave speed (v) can be related to the angular wave number (k) and angular frequency (ω) using the relationship:

k = 2π / λ

ω = 2π / T

Substituting the known values:

k = 2π / 14

ω = 2π / 1.5

Now we can write the equation for the wave:

y(x, t) = 3.6 * sin((2π / 14) * x - (2π / 1.5) * t)

This equation describes the sinusoidal ocean waves with a crest 14 m apart, and the surfer rising a distance of 3.6 m from the trough to crest in 1.5 s.

To find the wave speed, we can use the equation:

v = λ / T

Where:
v is the wave speed
λ (lambda) is the wavelength (distance between two consecutive crests or troughs)
T is the period of the wave (time required for one complete oscillation)

In this case, the wavelength is given as 14 m and the period is given as 1.5 s. We can substitute these values into the equation to find the wave speed:

v = 14 m / 1.5 s
v = 9.33 m/s

So, the wave speed is 9.33 m/s.

To describe the wave using an equation, we can use a general equation for a sinusoidal wave:

y(x, t) = A * sin(kx - ωt + φ)

Where:
y is the displacement of the wave at position x and time t
A is the amplitude of the wave (half the vertical distance between the crest and trough)
k is the wave number (related to wavelength by k = 2π / λ)
ω (omega) is the angular frequency of the wave (related to period by ω = 2π / T)
φ is the phase constant or phase shift

In this case, the amplitude is half the vertical distance between the crest and trough, which is (3.6 m / 2) = 1.8 m. The wave number can be calculated as k = 2π / λ = 2π / 14 m. The angular frequency can be calculated as ω = 2π / T = 2π / 1.5 s.

Substituting these values into the equation, we get:

y(x, t) = 1.8 * sin((2π / 14) * x - (2π / 1.5) * t + φ)

So, the equation that describes the wave is y(x, t) = 1.8 * sin((π / 7) * x - (4π / 3) * t + φ), where φ is the phase constant or phase shift.