Image copyright AP

The best of times, the worst of times? After a week in Cleveland, many Republicans don't know which.

There is a Trump wave, with power that everyone has come to respect, because it has swept away his rivals and created a mood. But his is also a presidential campaign that is a strange mixture of hope and incompetence.

Old convention hands are boggling at the mismanagement of this week, even as Trump believers say their conviction of a November win has been strengthened.

His wife's plagiarised speech gave rise to a series of Trump rebuttals that were a satirist's dream.

The cast of The Thick of It couldn't have made a better meal of it. The Ted Cruz stand-off was the kind of thing that campaigns are supposed to prevent at this stage. And the parade of Trump children and employees of "the organization" on the podium gave proceedings the feeling of a company away day.

Image copyright Reuters Image caption Hillary Clinton was the focus of much anger at the Republican Convention

Yet mockery from his opponents - of which there will be plenty in Philadelphia at the Democratic Convention next week - doesn't deal with the fact of Trump's popularity. National polls continue to give Clinton an edge, but not a lead that should give Democrats any feeling of comfort. People are listening to Trump.

And in his speech last night he found a tone that will serve him well when he addresses those that he's found to be angry, and anxious for an answer to their problems - alarm about immigration, fear of crime, the disappearance of jobs overseas and, above all, the feeling that the old promise of a guaranteed growth in prosperity has been betrayed.

To that, he says simply, "I am your voice." It is raw populism. The answers never come with detail. He says, "I'll fix it" or "Just watch me" and that's about it. But he has a feeling for the pulse of the Americans he calls "the disappointed." There are many of them.

But his speech will not reassure those who dislike his bombast and his tone. It's an appeal to emotion, and he was quite happy last night to stir up the crowd to join in their favourite chant - "lock her up!" - when he spoke of Hillary Clinton.

Image copyright Getty Images Image caption New Jersey Governor Chris Christie made a hard-hitting and controversial speech

She is battling against public disquiet about her admitted mistake in using a personal email server when she was secretary of state, but I met Republicans who were very uncomfortable at the sight of a plane circling over Cleveland throughout the week trailing a banner that read "Hillary for Prison 2016".

Governor Chris Christie's speech on Wednesday in which he invited the crowds to cry "Guilty!" half a dozen times in an imaginary courtroom was, for many people there, the crudest speech they had ever heard at a convention. For Democrats, it confirms the viciousness of the Trump campaign and their belief that, for him, anything goes.

But he's likely to moderate his tone when the campaign proper begins to pick up speed after Labor Day at the beginning of September. He knows that he has to balance his street fighter's instincts with the need to persuade people that he's not just a colourful and passionate candidate who speaks their language, but a president-in-waiting.

Image copyright Reuters Image caption Mr Trump's speech was an opportunity to reach beyond his core vote

Despite the likely bounce in the polls that he'll get from this week, that will require some work.

In his convention speech his commitments were so broad - on crime, immigration, trade, international relations, jobs - that he is going to be spending a great deal of time explaining precisely how he would go about delivering them. So far he's been able to stay afloat without being weighed down by detail, but that will change.

It's then that Donald Trump will be tested. Last night's speech was an important moment for him to present himself to the American people - for the first time in that form. The best guess, surely, is that it will bring him some new admirers.

But his next set-piece appearance on the national stage will be quite different. The first televised debate at the end of September is when street fighter and lawyer come face-to-face. On top of last night's rhetoric he will need to show substance, or he might find himself hitting the canvas.

How many candid friends he has got is hard to say. But they should be saying to him after this speech, "Don't get carried away. This is only the beginning."

James Naughtie is BBC News Book Editor and presents Bookclub on BBC Radio 4. He was a presenter on the Today programme from 1994 to 2015.