Solving Conduction Heat Wave in Metal Bar

[roam]

Homework Statement

A conduction heat wave is caused to pass through a metal bar of average temperature T_mean = 35.0 °C, altering its temperature sinusoidally with an amplitude of T_i = 2.0 °C. The equation which gives the temperature, T(x,t), of the metal bar at any place x (in meters) inside it at any time t (in seconds) is:

T(x,t) = T_mean + T_i sin[ 2π(0.027t – 3.0x) + Co ]

where C_o = 1.0π.

(a) What is the amplitude of the wave? (b) What is the wavelength of the wave?

Homework Equations

• Speed of sound wave (where B is the bulk modulus and mu p is the pressure): $$v=\sqrt{\frac{B}{\rho}}$$
  
• Power of sinusoidal wave: $$P=\frac{1}{2} \mu \omega^2 A^2 v$$
  
• For sound traveling through air:$$v=(331)\sqrt{1+\frac{T_C}{273}}$$

• Wave function for a sinusoidal wave: $$y=A sin(kx-\omega t)$$

The Attempt at a Solution

I don't understand how to apprach this problem. For example for part (a), what formula can I use? Also how do I find the temprature using the given equation:

T(x,t) = T_mean + T_i sin[ 2π(0.027t – 3.0x) + Co ]

what values do I need to substitute for "x" and "t"?

 

[diazona]

Also how do I find the temprature using the given equation:

T(x,t) = T_mean + T_i sin[ 2π(0.027t – 3.0x) + Co ]

Where does it ask you to find the temperature?

what values do I need to substitute for "x" and "t"?

The temperature is different at different positions and different times. To find the temperature at a specific point in space and at a specific time, you would substitute in the position coordinate (x) of that point in space, and the time coordinate t.

Part (a) asks you for the amplitude of the wave. Do you know what amplitude means? Do you know how to find it from a wave equation? For instance, the equation
$$y=A \sin(kx - \omega t)$$
represents a wave. What would be the amplitude of that wave?

 

[roam]

Part (a) asks you for the amplitude of the wave. Do you know what amplitude means? Do you know how to find it from a wave equation? For instance, the equation
$$y=A \sin(kx - \omega t)$$
represents a wave. What would be the amplitude of that wave?

"A" represents the amplitude. But how should I find the amplitude in this particular problem? The equation given only describes the temprature not the wave.

 

[diazona]

Well, you know that
$$y(x,t)=A \sin(kx - \omega t)$$
represents a wave, right? Do you also accept that
$$y(x,t) = y_\text{mean} + A \sin(kx - \omega t + \phi)$$
represents a wave? (How would you find its amplitude?)

If so, what's the problem? The equation
$$T(x,t) = T_\text{mean} + T_i \sin[2\pi(0.027t - 3.0x) + C_0]$$
is exactly the same thing, just with different letters.

 

[roam]

Thanks, I get it now.

Here's my last question: they further ask "What is the value of T(x,t) when t = 170.0 s, and x = 210.0 mm?"

I simply substituted the given values of "t" and "x" (along with other previously given values) into the equation:

T(x,t) = T_mean + T_i sin[ 2π(0.027t – 3.0x) + C_o]

But the value I got was not the correct answer. Why is that?

 

[Killeregg]

did u convert into metres?

 

[roam]

did u convert into metres?

40 mm = 0.04 m

$$37 sin(2 \pi (0.027(130)-3(0.04)+)\pi) = -23.5$$

But the correct answer must be 33.7! Why??