How to set and remove alarm set

Read your user's manual. Don't have it?
Download it from here

http://worldwide.bose.com/productsupport/assets/pdf/guides/wave/wrcd/en/owg_en_wrcd.pdf

Bose Support Owner Manuals

There's a setting to turn the touchpad on or off. Yours has probably been turned off. Here's how to turn it on.
1. Press and hold Alarm Setup (Menu) on your remote until -SETUP MENU- appears in the display.
2. Press Tune/MP3 until TOUCH PAD- is displayed (8 presses)
3. Press Time- or Time+ to select OFF or ON
4. Press Alarm Setup (Menu) to exit the menu, or simply wait 10 seconds.
You're done!

Random is truly a difficult problem to solve, if no pattern of failure can be detected. It sounds as though the alarm may be at fault. Is the alarm on and set to radio, CD, or beep? If radio, that could pinpoint the problem. If you have another, mechanical alarm, use it for a couple nights with the Bose Wave alarm off. If the Bose still acts up, then it isn't the alarm. Try unplugging the unit, waiting at least 30 seconds, and plugging it back in, which will cause you to need to reset the alarm. Alternately, remove the Bose cover and blow out dust. I find the Bose susceptible to dust as a cause of malfunction, which may point to the cause being the compact design of the unit. Good luck.

This is a picture of the Bose Radio Wave part of the manual that explains how to Turn Off the Second Alarm. And I added too the link for the page where you can find the original manual.
http://products.bose.com/pdf/customer_service/owners/wrv3_guide.pdf

Page 18

http://www.retrevo.com/support/Bose-Wave-radio-Radios-manual/id/525bh044/t/2/

To set AM/PM (12 hour) or Military (24 hour) time - Press the On/Off button once to turn your Wave radio off.

Press and hold Clock Set, and then press On/Off to switch between AM/PM (12 HR) or military (24 HR) formats

To set the Alarm 1 time - On the control panel, hold down ALARM
SET and press TUNING up or down. The time display changes slowly at
first, then more quickly. Release TUNING to stop.
On the remote, press TIME. The ALARM indicator flashes. Press
TUNING up or down. The time display changes slowly at first, then
more quickly. Release TUNING to stop. Press TIME to return the display
to the current time.
To turn on the Alarm 1 function - Press ALARM MODE or
ALARM in sequence to select the mode you want: tone, radio,
both tone and radio, or off.
For tone only, press until the indicator lights.
For radio only, press until the indicator lights.
For tone and radio, press until both indicators light. When the
alarm goes off, the tone sounds. Press SLEEP/SNOOZE to turn the
radio on. Press SLEEP/SNOOZE to pause the radio for 10 minutes.
To turn off the Alarm 1 function - Press ALARM MODE or
ALARM until the display shows no Alarm 1 mode indicators.
To check the Alarm 1 setting - On the control panel, hold down
ALARM SET to see the wakeup time.
On the remote, press TIME to see the wakeup time. Press TIME again to
return the display to current time.
To wake to a preselected radio station with Alarm 1 - While
holding ALARM SET, press the PRESET button for the station you want
(change between AM and FM if needed).

You can obtain the Bose Wave radio instruction manual in electronic form, here: http://products.bose.com/pdf/customer_service/owners/og_wrcd.pdf

See page 15 for setting the time. You should be able to do it with the unit controls.

http://products.bose.com/pdf/customer_service/owners/og_wrcd.pdf
follow this link to view the user manual for your radio and it includes all the instructions. refer to p. 20 of the manual for alarm setting instructions. If you are still having issue you may need to contact bose as your unit may be in need of repair.
Thank you,
Lee

Hold the alarm setup button and at the same time hit the on/off button.