Calculating Waves on a String: Answer Justification and Confusion Dispelled"

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bobrsanchez
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Hi all, I cannot seem to find a way to justify my book's answer to this problem:

One end of a string is shaken each second sending a wave with an amplitude of 10cm toward the other end. The string is 5m long and wavelength of each wave is 50cm. How many waves reach the other end of the string in each 10s interval?

Now my initial answer was calculated by figuring out the wave velocity = 0.5m/second using v=wavelength times frequency. I then divided the length of string by velocity to get 10 seconds for each wave to reach the other end of the string. Therefore my answer was 1 every 10 second interval.

The book however simply states that since frequency is 1 wave per second, then regardless of all other information given, the answer is that 10 waves reach the other end of the string every 10 seconds.

I find that answer very hard to swallow, mainly because for that to happen, that means the wave velocity must be 5m/s in order for each wave to only take one second to reach the other end, and I simply do not see that happening. Could somebody please dispel my confusion? Thanks in advance!

Edit: I think I see what my book is saying now. But in order for that to be the case, the string must be saturated with waves to begin with, right? In other words, wouldn't the very first interval of 10 seconds still only have 1 wave reaching the other end of the string?
 
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Now my initial answer was calculated by figuring out the wave velocity = 0.5m/second using v=wavelength times frequency. I then divided the length of string by velocity to get 10 seconds for each wave to reach the other end of the string. Therefore my answer was 1 every 10 second interval.

You are forgetting that there is more than one wave on the string at anyone time. Your answer assumes that one wave is launched (taking 1 second) then the man waits 9 seconds before launching another.

The book answer is correct. In 10 seconds 10 waves are launched so 10 must arrive at the other end.

Edit: I think I see what my book is saying now. But in order for that to be the case, the string must be saturated with waves to begin with, right? In other words, wouldn't the very first interval of 10 seconds still only have 1 wave reaching the other end of the string?

That's correct.

The question says "How many waves reach the other end of the string in each 10s interval?" So a full answer would be something like "In the first 10 seconds only 1 wave reaches the other end. In the next and all subsequent 10 second intervals 10 waves reach the other end".
 
Okay thank you very much. Thats what I suspected to be the long answer the book neglected to provide. Thanks again for confirming!

Cheers