Description of a wave

  • #1

Homework Statement


A simple harmonic wave train of amplitude 3 cm and frequency 200 Hz travels in the +ve direction of x-axis with a velocity of 20 m/s. Calculate the displacement, velocity, and acceleration of a particle situated at 50 cm from the origin at t = 2 s.

Homework Equations


I used [itex]y(x, t) = Acos(2\pi f(\frac{x}{v}-t))[/itex]

The Attempt at a Solution


Plugging in the values into the above equation, I got [itex]y(0.5, 2) = 0.03cos(400\pi (\frac{.5}{20} - 2))[/itex], which evaluates to 0.0236 m. However, the book says the answer is 0.02523 m.

This is marked as an easy question and is one of the first ones, so I think that I'm missing something basic?
 

Answers and Replies

  • #2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,415
6,945
You need to be a bit careful evaluating trig functions of large angles. The series expansions used by calculators get rather inaccurate. Instead, first reduce the angle to something less than 2 pi. I think you'll find both your answer and the given answer rather inaccurate!
 
  • #3
Part of my confusion lies in whether I should be in radians or degrees - the examples in the book all use radians, but when I evaluated the above, I got [itex]cos(-790\pi)[/itex], or 1. Changing the angle to something less than 2[itex]\pi[/itex] still gets me 1.

When I use degrees, I get 0.0236, which is closer to 0.02523. I plugged the expression into Wolfram-Alpha, which got me the same as the one on my calculator. Changing the angle to something less than 360 degrees got me 0.0271.
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,415
6,945
Part of my confusion lies in whether I should be in radians or degrees - the examples in the book all use radians, but when I evaluated the above, I got [itex]cos(-790\pi)[/itex], or 1.
Definitely radians. The standard equation you quoted assumes radians.
It gives you 1 for the value of the cos function, but you still have to multiply by A.
 
  • #5
rude man
Homework Helper
Insights Author
Gold Member
7,931
822

Homework Statement


A simple harmonic wave train of amplitude 3 cm and frequency 200 Hz travels in the +ve direction of x-axis with a velocity of 20 m/s. Calculate the displacement, velocity, and acceleration of a particle situated at 50 cm from the origin at t = 2 s.

Homework Equations


I used [itex]y(x, t) = Acos(2\pi f(\frac{x}{v}-t))[/itex]

The Attempt at a Solution


Plugging in the values into the above equation, I got [itex]y(0.5, 2) = 0.03cos(400\pi (\frac{.5}{20} - 2))[/itex], which evaluates to 0.0236 m. However, the book says the answer is 0.02523 m.

This is marked as an easy question and is one of the first ones, so I think that I'm missing something basic?
I got the same answer as you.
BTW the problem should state that the wave is inded ~ cos(kx - wt) and not something like cos(kx - wt + φ), φ ≠ 0.
Part of my confusion lies in whether I should be in radians or degrees
Always assume radians. And always assume natural instead of base-10 logs. Calculus falls apart otherwise!
 
  • #6
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,415
6,945
I got the same answer as you.
Then you must be using a calculator that truncates the precision of pi at the same point. The right answer is clearly 0.03m.
 
  • #7
rude man
Homework Helper
Insights Author
Gold Member
7,931
822
Then you must be using a calculator that truncates the precision of pi at the same point. The right answer is clearly 0.03m.
Not truncate. Round off.
But yes, score one for the Aussies. Again, only if the wave is cos(kx - wt + φ), φ = 0 assumed. The problem is not clearly stated.
 

Related Threads on Description of a wave

  • Last Post
Replies
1
Views
9K
  • Last Post
2
Replies
30
Views
4K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
1
Views
547
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
8
Views
581
Replies
6
Views
3K
  • Last Post
Replies
1
Views
773
Top