Time Machine "the backup disk could not be found" error

Simple first step: restart your Mac in the usual manner.

If that does not fix it, "reconnect" to the backup disk using the procedure described below.

The following technique can be used to "reconnect" to the backup disk. It will not affect the existing backup history in any way.

Open Time Machine's Preferences and de-select the Back Up Automatically checkbox. Make sure no backups are in progress, otherwise you should wait for it to finish or terminate.

• If the backup disk is directly connected to the Mac, disconnect it: drag its icon to the Trash, wait for it to disappear from the Desktop, and then physically disconnect it from the Mac.
• If its icon does not appear on the Desktop, select External Disks in the Finder menu > Preferences... > General. You can de-select it later.

Then, click the "Select Disk..." button, select the backup disk, then Remove Disk, then Stop Using This Disk.

• If you designated more than one backup destination, the "Select Disk..." button is replaced by an "Add or Remove Backup Disk..." option. You may have to scroll through the list of backup disks for that option to become visible.

This will not affect any existing backups.

• If the backup disk had been directly connected to your Mac, reconnect the backup disk, power it on if required, and wait for its icon to appear on the Desktop.

Then, Select Disk... again in Time Machine's Preferences. Re-select the same one under Available Disks, then Use Disk.

• If the backup is on a network (TC or AEBS), you will be prompted for the password you provided for that disk in AirPort Utility.
• If the backup was encrypted, you will also be asked for the backup encryption password.

"Waiting to complete first backup" will appear, which means it won't do anything until the next scheduled backup. Despite the implication of that message, it does not mean it needs to create a completely new backup, as if it's creating a brand new one. Your existing backups will still be available.

• If you don't want to wait that long, select Back Up Now.

A Notification may appear if your source volume is encrypted and the backup volume is not. That's normal.

Re-select Back Up Automatically. The "Preparing Backup..." status message will remain for a long time. Eventually, "Backing up xxx of yyy" will appear, but those values will not be accurate, nor will the "Estimated time remaining" in Time Machine's Preferences. Just ignore it. It may take a few hours even if your Mac is not allowed to sleep. Time Machine will also pause or become slow if you use your Mac for tasks it considers a higher priority.