A conduction heat wave is caused to pass through a metal bar of average temperature Tmean = 36 degrees altering its temperature sinusoidally with an amplitude of Ti=2 degrees. The equation which gives the temperature T(x.t) of the metal bar at any place x(in metres) inside it at any time t (in seconds) is : T(x,t) = Tmean+Ti sin[2£k(0.029t ¡V 9.0x)+ 2£k]
a)What is the wavelength of the wave?
b)What is the period of the wave?
I can't read the equation.
I think its meant to be T (x,t) = 36 + 2 sin[2pie(0.029t - 9x) + 2pie]
In that case, one wavelength occurs at 2PI=9x or x= 2/9 PI
To find the wavelength and period of the wave described by the equation T(x,t) = Tmean + Ti sin[2πk(0.029t – 9.0x) + 2πk], we can use the formulae for wavelength and period in a sinusoidal function.
a) Wavelength:
The wavelength (λ) is the distance between two consecutive points in the wave that have the same phase. In this case, the wave is described by the term 2πk(0.029t – 9.0x) + 2πk within the sine function.
To find the wavelength, we need to consider the argument of the sine function, which is 2πk(0.029t – 9.0x) + 2πk. The argument corresponds to the phase of the wave.
Since the cosine function has a period of 2π, the argument of the sine function should increase by 2π over the distance of one wavelength. Therefore, to find the wavelength, we equate the argument to 2π and solve for the distance x.
2πk(0.029t – 9.0x) + 2πk = 2π
Simplifying the equation:
0.029t – 9.0x = 1
Now, we can solve for x by isolating it:
-9.0x = 1 - 0.029t
x = (1 - 0.029t) / -9.0
From this equation, we can see that the coefficient of t (-0.029) can be considered as the inverse of the wavelength (1/λ). So, the wavelength (λ) is equal to 1 / (-0.029), which simplifies to approximately -34.48 meters. However, for practical purposes, we take the absolute value, so the wavelength is approximately 34.48 meters.
b) Period:
The period (T) of a wave is the time it takes for one complete cycle. It can be calculated using the formula T = 2π / ω, where ω is the angular frequency.
In this case, the angular frequency is given by the coefficient of t in the argument of the sine function, which is 2πk(0.029t – 9.0x) + 2πk. Therefore, the angular frequency is 0.029.
Using the formula for period, we have:
T = 2π / ω
T = 2π / 0.029
T ≈ 216.19 seconds
So, the period of the wave is approximately 216.19 seconds.